evaluating multi-horizon volatility-forecasting ... · volatility forecasting of financial assets...

Master’s Thesis Financial Economics Erasmus School of Economics, Erasmus University Rotterdam

Evaluating Multi-Horizon Volatility-Forecasting Performances of GARCH-Type Models

Author: Han-Ywan Hu

ERNA no.: 449758

Supervisor: Dr Rogier Quaedvlieg

Second assessor: Dr Esad Smajlbegovic

Date: 7 February 2019

Acknowledgements

I would like to express my sincere gratitude to Dr Rogier Quaedvlieg, my thesis supervisor, for his

eternal patience, guidance, useful comments, and flexibility to let me pursue two banking internships

at ABN AMRO and ING during my study. Besides, I would like to thank the second assessor for his

courage to read this tedious paper. Furthermore, I would also like to thank my family, my dearest

friends, and everyone else for their encouragement and support during this prolonged but rewarding

journey. This accomplishment would have not been possible without them.

万事开头难。

冰冻三尺，非一日之寒。

Rotterdam, 7 February 2019

Han-Ywan Hu

NON-PLAGIARISM STATEMENT By submitting this thesis the author declares to have written this thesis completely by himself/herself, and not to have used sources or resources other than the ones mentioned. All sources used, quotes and citations that were literally taken from publications, or that were in close accordance with the meaning of those publications, are indicated as such.

COPYRIGHT STATEMENT The author has copyright of this thesis, but also acknowledges the intellectual copyright of contributions made by the thesis supervisor, which may include important research ideas and data. Author and thesis supervisor will have made clear agree-ments about issues such as confidentiality.

Electronic versions of the thesis are in principle available for inclusion in any EUR thesis database and repository, such as the Master’s Thesis Repository of the Erasmus University Rotterdam


Han-Ywan Hu

Erasmus School of Economics, Erasmus University Rotterdam

7 February 2019

Abstract

This paper examines the forecast performances of several GARCH models at different horizons for

equity indices from developed markets and emerging markets. It tries to find out which GARCH mod-

els are best for short-term volatility forecasts and which ones perform best at longer horizons. Fur-

thermore, we also study the underlying features that drive the forecast performance of GARCH mod-

els at multiple horizons. We use the ARCH, GARCH, GJR-GARCH, CGARCH, APARCH, and GARCHX mod-

els to forecast out-of-sample conditional variances of S&P 500, HSI, IPC, and KOSPI. The forecast per-

formances are evaluated with MSE and QLIKE. Additionally, we test the significance of forecast losses

with the Diebold-Mariano test and Model Confidence Set (MCS). The empirical findings of this paper

are mixed, since there is not a single model that shows superior performance in each case. The

GARCHX model, which incorporates the VIX, tends to generate superior forecasts at multiple horizons

for the S&P 500, whereas in most cases CGARCH and APARCH are preferred under the remaining

indices. Overall, we can conclude that conjoining a precise volatility proxy of an index to an ordinary

GARCH model significantly improves the forecast performance, yet there are other factors as well that

may influence the predictive ability of models.

Keywords: Forecasting, ARCH, GARCH, GJR-GARCH, CGARCH, APARCH, GARCHX, Realised Volatility, VIX

JEL: C22, C52, C53, C58

Table of contents

§1. Introduction ...................................................................................................................................................... 1

§2. Literature ........................................................................................................................................................... 3

2.1. Multi-horizon forecasting................................................................................................................................................................ 3

2.2. Empirical regularities of asset return volatility .................................................................................................................... 4

• 2.2.1. Fat-tailed distributions .......................................................................................................................................................... 4

• 2.2.2. Volatility clustering ................................................................................................................................................................. 4

• 2.2.3. Long-memory property.......................................................................................................................................................... 4

• 2.2.4. Leverage effect .......................................................................................................................................................................... 5

• 2.2.5. Volatility-spillover effect ....................................................................................................................................................... 5

• 2.2.6. Commonality in volatility changes .................................................................................................................................... 5

§3. Methodology ..................................................................................................................................................... 6

3.1. Volatility proxies ................................................................................................................................................................................. 6

• 3.1.1. Squared return .......................................................................................................................................................................... 6

• 3.1.2. Realised variance ..................................................................................................................................................................... 7

• 3.1.3. Range-based variance ............................................................................................................................................................ 8

3.2. Forecasting models ............................................................................................................................................................................ 9

• 3.2.1. ARCH ............................................................................................................................................................................................. 9

• 3.2.2. GARCH ....................................................................................................................................................................................... 10

• 3.2.3. GJR-GARCH ............................................................................................................................................................................... 11

• 3.2.4. CGARCH ..................................................................................................................................................................................... 11

• 3.2.5. APARCH ..................................................................................................................................................................................... 12

• 3.2.6. GARCHX ..................................................................................................................................................................................... 13

3.3. Model fitting & forecast evaluation .......................................................................................................................................... 14

• 3.3.1. Information criteria tests .................................................................................................................................................. 14

• 3.3.2. Mean squared error ............................................................................................................................................................. 14

• 3.3.3. Quasi-likelihood ..................................................................................................................................................................... 15

• 3.3.4. Diebold-Mariano test ........................................................................................................................................................... 16

• 3.3.5. Model Confidence Set ........................................................................................................................................................... 17

3.4. Forecasting procedure ................................................................................................................................................................... 18

§4. Data .................................................................................................................................................................... 20

4.1. Descriptive statistics ...................................................................................................................................................................... 21

§5. Empirical results ........................................................................................................................................... 29

5.1. In-sample fitting results ................................................................................................................................................................ 29

5.2. Forecast evaluation ......................................................................................................................................................................... 30

• 5.2.1. Results of loss functions ...................................................................................................................................................... 30

• 5.2.2. Results of Diebold-Mariano tests .................................................................................................................................... 36

Table of contents

• 5.2.3. Results of MCS ......................................................................................................................................................................... 36

§6. Research limitations .................................................................................................................................... 39

§7. Suggestions for further research ............................................................................................................. 40

§8. Summary and conclusion ........................................................................................................................... 41

References .............................................................................................................................................................. 44

Appendix ................................................................................................................................................................. 49

Section A – Significance tests .............................................................................................................................................................. 49

• A.1. Jarque-Bera normality test ................................................................................................................................................... 49

• A.2. Augmented Dickey-Fuller test ............................................................................................................................................. 49

• A.3. Ljung-Box portmanteau test ................................................................................................................................................ 51

• A.4. Engle’s ARCH test...................................................................................................................................................................... 52

Section B – Supplementary tables ..................................................................................................................................................... 54

Section C – Supplementary graphs ................................................................................................................................................... 66

List of tables

Table 1. – Data sample ................................................................................................................................................................................. 19 Table 2. – List of stock indices .................................................................................................................................................................. 20 Table 3. – Summary statistics of stock index returns .................................................................................................................... 21 Table 4. – Correlations of stock indices returns ............................................................................................................................... 21 Table 5. – Summary statistics of volatility proxies ......................................................................................................................... 23 Table 6. – Results of in-sample goodness-of-fit tests ..................................................................................................................... 29 Table 7. – Forecast loss coefficients of S&P 500 .............................................................................................................................. 31 Table 8. – Summary of forecast models with the lowest loss coefficients ........................................................................... 32 Table 9. – Average forecast losses .......................................................................................................................................................... 33 Table 10. – Results of MCS for S&P 500 ............................................................................................................................................... 37 Table A1. – Unit root tests for price levels and log returns ......................................................................................................... 50 Table A2. – Unit root tests for volatility proxies .............................................................................................................................. 51 Table A3. – Ljung-Box test statistics ...................................................................................................................................................... 52 Table A4. – ARCH LM test statistics ....................................................................................................................................................... 53 Table B1. – In-sample estimates of model parameters ................................................................................................................. 54 Table B2. – Forecast loss coefficients of HSI ...................................................................................................................................... 56 Table B3. – Forecast loss coefficients of IPC ...................................................................................................................................... 57 Table B4. – Forecast loss coefficients of KOSPI ................................................................................................................................ 58 Table B5. – Coefficients of determination ........................................................................................................................................... 58 Table B6. – Diebold-Mariano test statistics of one-day ahead forecasts of S&P 500 ....................................................... 59 Table B7. – Diebold-Mariano test statistics of one-week ahead forecasts of S&P 500 ................................................... 60 Table B8. – Diebold-Mariano test statistics of one-month ahead forecasts of S&P 500 ................................................. 61 Table B9. – Diebold-Mariano test statistics of one-quarter ahead forecasts of S&P 500 .............................................. 62 Table B10. – Results of MCS for HSI....................................................................................................................................................... 63 Table B11. – Results of MCS for IPC ....................................................................................................................................................... 64 Table B12. – Results of MCS for KOSPI ................................................................................................................................................. 65

List of figures

Figure 1. – Kernel density plots of stock index returns ................................................................................................................ 22 Figure 2. – Annualised volatility proxies of S&P 500 index ........................................................................................................ 24 Figure 3. – Autocorrelations of S&P 500 volatility proxies ......................................................................................................... 25 Figure 4. – Partial autocorrelations of S&P 500 volatility proxies ........................................................................................... 26 Figure 5. – Historical behaviour of the VIX Index ............................................................................................................................ 27 Figure 6. – S&P 500 index .......................................................................................................................................................................... 28 Figure 7. – One-step ahead volatility forecasts of S&P 500 ......................................................................................................... 34 Figure 8. – One-step ahead volatility forecasts of S&P 500 ......................................................................................................... 35 Figure C1. – Annualised volatility proxies of HSI index ................................................................................................................ 66 Figure C2. – Annualised volatility proxies of IPC index ................................................................................................................ 67 Figure C3. – Annualised volatility proxies of KOSPI index .......................................................................................................... 68 Figure C4. – Autocorrelations of HSI volatility proxies ................................................................................................................. 69 Figure C5. – Autocorrelations of IPC volatility proxies ................................................................................................................. 70 Figure C6. – Autocorrelations of KOSPI volatility proxies ........................................................................................................... 71 Figure C7. – Partial autocorrelations of HSI volatility proxies .................................................................................................. 72 Figure C8. – Partial autocorrelations of IPC volatility proxies................................................................................................... 73 Figure C9. – Partial autocorrelations of KOSPI volatility proxies............................................................................................. 74 Figure C10. – HSI index ............................................................................................................................................................................... 75 Figure C11. – IPC index ................................................................................................................................................................................ 76 Figure C12. – KOSPI index .......................................................................................................................................................................... 77 Figure C13. – Historical behaviour of variance proxies of S&P 500 ........................................................................................ 78

Master’s Thesis Financial Economics H. Hu

1

§1. Introduction

“We really can’t forecast all that well, and yet we pretend that we can, but we really can’t.”

─ Alan Greenspan

Volatility forecasting of financial assets such as bonds and equities is a common practice for scholars

and practitioners in the field of corporate finance.1 As mentioned in Poon and Granger (2005), vola-

tility forecasting has a significant role, for instance, in risk management, investment analysis, and

pricing of derivatives. Especially in option pricing theory, the volatility of an underlying asset is one

of the key parameters in the notorious Black-Scholes-Merton formula, where one can determine the

theoretical value of an option contract (Figlewski, 1997). Besides its essential role in option pricing,

it also has a key role in risk management where volatility is an element of Value-at-Risk models (Hull,

2012). Undoubtedly, finance professionals are faced with different types of financial risks on a daily

basis. Therefore, a forward-looking volatility measure of asset prices over a holding period can be

considered as a good starting point for evaluating investment risk (Poon & Granger, 2003).

From an academic perspective, many studies have conducted research on forecasting models and

their predictive abilities. In addition, studies have directly compared the forecasting performances of

economic variables of these forecasting models. For example, Poon and Granger (2003) delineate the

results of forecast performances of various volatility-forecasting models from 66 studies. However, a

sceptical reader may question the validity of mathematical and statistical models for forecasting pur-

poses. Firstly, can we predict future movements of economic variables such as volatility of a financial

asset exactly? Secondly, which model is considered as the best model for volatility forecasting? The

first question sounds rather naïve since economists often regard the future as uncertain, which of

course complicates the matter of making a precise prediction of the future. According to the renowned

essayist Nassim Nicholas Taleb, extreme events such as the 9/11 terrorist attacks in the U.S. are un-

known unknowns.2 These rare events defy any prediction made by statistical models because these

events are one-off (Kay, 2007). Thus, future movements of economic variables are unlikely to be pre-

dicted exactly by existing forecast models. Then, the answer to the second question is rather ambigu-

ous due to the mixed results of forecast performances reported in previous studies, which will be

explained in more detail in the next section. Moreover, asset returns often do not contain sufficient

1 In this paper, the words volatility, 𝜎, and variance, 𝜎2, are interchangeable. This should not lead to conceptual confusion since these two statistical measures are related to each other through a monotonic transformation. 2 Taleb labels unknown unknowns as “black swans”.


2

information to pinpoint a generic forecast model as “best” (Hansen et al., 2003). Therefore, it is hard

to find a clear winner when it comes to forecasting economic variables.

The global financial crisis of 2007-2008 clearly revealed that heavy reliance on complex mathemat-

ical and statistical models do not necessary lead to correct predictions of future market movements.

Despite the frequent criticism of forecasting models, it is important to note that economic and finan-

cial outlooks are necessary for executives, institutional investors, investment bankers, and policy-

makers because they need to anticipate certain shocks. They need mathematical and statistical tools

to understand the nature of uncertainty to make well-informed decisions. Therefore, many mathe-

matical models are still used for forecasting purposes nowadays.

The aim of this paper is mainly to evaluate the forecast performance of several GARCH-type models

at different horizons, also known as multi-horizon forecasting. Further, it studies the underlying fea-

tures that drive the forecasting performance of these GARCH models at multiple horizons. Based on

this, the following research questions can be formulated: (1) “Which of the following GARCH-type mod-

els generate the best volatility forecasts in the short term: ARCH, GARCH, GJR-GARCH, CGARCH, APARCH,

or GARCHX?”, (2) “Which of the following GARCH-type models generate the best volatility forecasts in

the medium term: ARCH, GARCH, GJR-GARCH, CGARCH, APARCH, or GARCHX?”, and (3) “What key fea-

tures drive the forecast performance of these GARCH-type models at multiple horizons?” The abovemen-

tioned research questions can be answered through evaluations of forecast results of these models at

multiple horizons, and by using several goodness-of-fit measures to determine the model’s signifi-

cance and predictive ability. Further, this study uses equity indices from both developed markets and

emerging markets.

Why is it worth to examine the abovementioned research questions? Firstly, from institutional in-

vestors’ perspective, it is important to know which forecast models are best applicable to different

types of financial assets or even within the same asset class. Secondly, it is important to examine and

compare the predictive abilities of both simple and sophisticated models. This paper studies the vol-

atility-forecasting performances and abilities of the ARCH, GARCH, GJR-GARCH, CGARCH, APARCH,

and GARCHX models. Finally, this study adds to existing literature by comparing the volatility-fore-

casting models’ performances of developed and emerging markets equity indices over horizons rang-

ing from one day to one quarter. Furthermore, it examines the underlying features of these GARCH-

type models that drive the forecast performance at different horizons.


3

§2. Literature

Previous studies related to multistep forecasting will be examined in this section, thus providing us

with more insights on volatility forecasting at different horizons and the well-known empirical regu-

larities of asset return volatility.

2.1. Multi-horizon forecasting

Throughout the years, numerous studies have done research on the use of distinct forecast models

and their ability to generate multistep forecasts of economic variables. For example, Marcellino et al.

(2006) compared out-of-sample forecasts of 170 different macroeconomic variables at multiple ho-

rizons, meanwhile Clark and McCracken (2005) focused their empirical study on a single macroeco-

nomic variable, namely core CPI inflation. Other studies like Ghysels et al. (2009), Brownlees et al.

(2011) and Quaedvlieg (2018) focused their research on volatility forecasting of securities by using

various forecast models. Moreover, Brownlees et al. (2011) also investigated how different forecast-

ing procedures, e.g. forecast window length, innovation distribution, and frequency of model param-

eter re-estimations, could affect the predictive ability of those GARCH models.

Surprisingly, consensus regarding which forecast model performs the best has not been reached

yet. For instance, Hansen and Lunde (2005) analysed more than 300 GARCH-type models in terms of

their ability to predict conditional variances of the Deutsche Mark-USD currency rate and IBM stock

returns. According to the authors, specifying the standard GARCH(1,1) model into a more complex

form does not necessarily lead to outperformance compared to the standard form. However, as re-

ported in the extensive literature review by Poon and Granger (2003), some empirical studies have

demonstrated evidence that even a simple historical volatility model, e.g. historical moving average

or random walk model, can beat a complex GARCH-type model.

There are two common methods to conduct multistep forecasts. One could use either the direct or

the iterated approach, which is also called recursive forecast. The former uses a horizon-specific fore-

cast model in which the response variable is the ℎ-steps ahead direct forecast at a certain time,

whereas the latter uses one-step ahead forecasts of a model to determine the ℎ-steps ahead forecast

recursively (Marcellino et al., 2006; Ghysels et al., 2009). In other words, ℎ-steps ahead iterated fore-

casts are based on a function of lagged one-step ahead forecasts. Interestingly, some previous studies

have demonstrated that multistep iterated forecasts tend to outperform direct forecasts in terms of

forecast accuracy, although improvements are relatively small for some horizons, see Marcellino et

al. (2006), Proietti (2011), and McCracken and McGillicuddy (2017).


4

2.2. Empirical regularities of asset return volatility

Finance literature has documented several important empirical regularities regarding historical as-

set returns and market volatility. For example, one of the characteristics is that asset return volatility

tends to persist and cluster (Poon & Granger, 2005). Another important characteristic is the asym-

metric nature of asset return volatility. The next subsubsections will briefly introduce some of these

empirical regularities from previous literature.

• 2.2.1. Fat-tailed distributions

As documented in Mandelbrot (1963), Fama (1965), and Bollerslev et al. (1994), asset returns tend

to be distributed in a leptokurtic way, i.e. a return distribution with positive excess kurtosis that con-

tains more observations in the extreme tails (Tsay, 2005). Similar results for stock index returns can

be found in Figure 1 (see Section 4.1) of this paper. This empirical regularity is also known as fat-

tailed distribution, or heavy tails, in finance literature.

• 2.2.2. Volatility clustering

Other empirical regularities are clustering of volatility and persistence of asset returns (Poon &

Granger, 2005). An example of volatility clustering is provided in Figure 2, where periods of relatively

high volatility tend to cluster with each other, and low-volatility periods tend to be trailed by periods

of relatively low volatility (Mandelbrot, 1963). In other words, there are cycles of substantial disper-

sion in asset returns and cycles of relative tranquillity. Although the figure illustrates hefty shocks in

asset return volatility over time, volatility tends to be mean reverting in general (Engle & Patton,

2001). According to Engle (1993), if substantial adjustments in markets tend to be tracked by similar-

sized changes, in either direction, then volatility must be predictably high after hefty changes.

• 2.2.3. Long-memory property

Persistence in volatility, or long memory, means that volatility shocks tend to decay slowly over

time. In other words, volatility is said to be persistent if today’s asset return not only has a significant

impact on conditional variance forecast of the next period, but also on the variance estimates many

periods in the future (Engle & Patton, 2001). Ding et al. (1993) and Ding and Granger (1996) have

demonstrated that the long-memory property exists for some speculative asset returns, such as the

S&P 500 and the DM/USD currency pair. Although speculative asset returns hold slight autocorrela-

tion, the absolute returns and their power transformations tend to be serially correlated (Conrad et

al., 2011). To detect this phenomenon, autocorrelations of variance estimates can be used to figure

out the persistence level of volatility. An example of this is demonstrated in Figure 3.


5

• 2.2.4. Leverage effect

The leverage effect refers to the tendency for asset price adjustments to be inversely correlated

with asset volatility changes (Bollerslev et al., 1994; Andersen et al., 2006). In other words, declining

asset prices are often accompanied by rising volatility, and vice versa (Aït-Sahalia et al., 2013). The

impact of asset returns on volatility tend to be relatively weaker in bull markets than in bear markets,

i.e. volatility is asymmetric over time (Poon & Granger, 2005). A common explanation for this effect

is the change in a company’s financial leverage, or its debt-to-equity ratio. In case when the enterprise

value of a firm falls, the company with debt and equity outstanding typically becomes mechanically

more levered (Bollerslev et al., 1994; Aït-Sahalia et al., 2013). Yet, the magnitude of the impact of a

stock price drop on future volatilities seems too substantial to be explained merely by adjustments in

company’s financial leverage (Bollerslev et al., 2006; French et al., 1987). Still, according to Christie

(1982), the inverted relationship between volatility and stock price is to a substantial degree attribut-

able to financial leverage. Christie (1982), Bouchaud et al. (2001), and Aït-Sahalia et al. (2013) have

tested the leverage effect empirically.

• 2.2.5. Volatility-spillover effect

In the past decades, globalisation of financial markets and financial integration have led to inter-

twined capital markets on a massive scale. Nowadays, global markets are becoming increasingly in-

terdependent, meaning that information flows instantaneously from one market to another, and thus

affecting the securities market abroad. This phenomenon is also known as volatility spillover in fi-

nance literature. Previous studies have documented volatility-spillover effects in distinct financial

markets. For instance, Hamao et al. (1990), Susmel and Engle (1994), Koutmos and Booth (1995), Ng

(2000), and Baele (2005) studied the price volatility spillovers between various stock markets, mean-

while Skintzi and Refenes (2006) and Christiansen (2007) conducted their research mainly on the

European bond markets. In addition, Tse (1999) and Yarovaya et al. (2017) investigated several stock

index futures markets for spillover effects.

• 2.2.6. Commonality in volatility changes

Another empirical regularity that is widely reported is the commonality in volatility changes, or the

comovements in asset return volatility across distinct financial assets and even across international

financial markets (Bollerslev et al., 1994). For example, changes in the volatility of equities and bonds

tend to comove in the same direction (Schwert, 1989). This empirical regularity is closely linked with

volatility-spillover effects.


6

§3. Methodology

3.1. Volatility proxies

A common issue in volatility forecasting is that conditional variance is not directly observable,

which complicates the evaluation and comparison of predictions (Andersen & Bollerslev, 1998). For-

tunately, there are ways to resolve this issue partially if an unbiased estimator of variance is available

(Patton, 2011b). Empirical studies have proposed numerous proxies for conditional variance such as

squared return, realised variance, and range-based variance, also called high-low range. Nevertheless,

using a conditionally unbiased variance proxy does not necessarily result in the same outcome as if

the true variance were used, as volatility proxies tend to be rather noisy in practice. Moreover, previ-

ous studies, such as Andersen et al. (1999) and Poon and Granger (2003), noted that a few outliers in

sample data might substantially affect the outcomes of forecast evaluation and comparison tests. One

way to tackle this issue, for example, is to use loss functions that are relatively less sensitive to outli-

ers. However, Patton (2011b) found that many of those loss functions are not robust and thus can

potentially result in erroneous rankings of conditional variance estimates. Throughout the years, no

clear consensus has been reached yet on the treatment of outliers by academics (Poon & Granger,

2003).

This paper uses realised variance and range-based variance as volatility proxies. These estimators

are described in the following subsubsections. The reason why we use these volatility proxies is be-

cause of forecast evaluation later. Before we move on to these subsubsections, we will first briefly

discuss about the squared return.

• 3.1.1. Squared return

According to Patton (2011a), the daily squared return of a financial asset is considered as an unbi-

ased volatility proxy that is simple and widely available for investors and researchers. However, de-

spite the convenience of this proxy measure, squared return is also considered as a very noisy indi-

cator of volatility (Andersen et al., 2001). While squared return is an unbiased estimator for the latent

variable of interest, the idiosyncratic error term causes the inaccuracy of this proxy (Andersen &

Bollerslev, 1998). As demonstrated in Lopez (2001), the squared error, 휀𝑡2, is 50% greater or smaller

than the conditional variance, 𝜎𝑡2, for nearly three-fourths of the time. As a result, the squared return

approach often leads to severe volatility forecasts. Hence, this method will not be used in this paper

as it is expected to underperform other variance proxies.


7

• 3.1.2. Realised variance

An alternative proxy for volatility is realised variance, or realised volatility (henceforth RV), which

is based on intraday returns. The availability of high-frequency data led to better estimates of daily

volatility, due to increased number of intraday observations. As described in Christoffersen (2012)

and Patton (2011b), the realised variance is simply the sum of all the intraday squared log returns.

Mathematically, it can be written as

𝑅𝑉𝑡𝑚 = ∑ 𝑟𝑡−1+𝑖 𝑚⁄

2

𝑚

𝑖=1

, (1)

where 𝑚 is the amount of observations available within a day assuming that those observations are

equally spaced, and 𝑖 denotes the 𝑖th intraday return on day 𝑡. Contrary to the daily squared return,

RV is a proxy that can illustrate the daily variance in multiple ways through changes in the time inter-

val, i.e. one can determine realised variance by using intraday observations that are sampled at, for

instance, 𝑘-minute or hourly intervals. As a result, numerous estimates of RV are available. According

to Poon and Granger (2005), from all the available RV estimates, studies have demonstrated that 5-

minute or 15-minute interval produces the best results. This paper uses 5-minute and 10-minute re-

alised volatilities since these time series are retrievable from Oxford-Man Institute’s realised library.

Like the daily squared return, RV is a valid proxy of the daily conditional variance (Patton, 2011b).

What characterises this volatility estimator is that in absence of market microstructure noise it is a

more efficient estimator than the daily squared return (Martens & van Dijk, 2007). Moreover, RV is

extremely persistent, which implies that volatility may be forecastable at short horizons if the infor-

mation in intraday returns is used (Christoffersen, 2012). According to Andersen et al. (2006), the

key feature of RV is that it provides a consistent nonparametric estimation of the variability of price

that has occurred over a given discrete interval.

Unfortunately, the presence of market microstructure frictions significantly affects the estimation

of realised variance. For example, in a univariate case, a bounce in the bid-ask spread of a financial

asset tends to increase the measured volatility of the asset’s high-frequency intraday returns. As a

result, the volatility proxy under Eq. (1) tends to be upwardly biased due to the cumulated bias in

intraday returns (Alizadeh et al., 2002). Market microstructure noise causes serial correlation in the

intraday returns and therefore makes the RV estimator biased (Hansen & Lunde, 2006). To diminish

the impact of market microstructure noise on volatility estimate, RV estimator can be constructed at

a moderate frequency. However, according to Christensen and Podolskij (2007), this leads to loss of

information.


8

• 3.1.3. Range-based variance

In presence of market microstructure noise, the construction of periodic realised volatilities be-

comes an arduous task, for example, due to inconsistent or lack of clean high-frequency data. In con-

trast to RV estimator, the range-based variance proxy, or range proxy (henceforth RP), is relatively

less susceptible to market microstructure frictions (Christoffersen, 2012). This estimator reveals rel-

atively more information about the actual volatility than intraday returns sampled at fixed intervals,

since the daily range values are formed from the entire price process (Christensen & Podolskij, 2007).

Similar to Christoffersen (2012), assuming that log returns are normally distributed with mean and

variance equal to zero and 𝜎2, respectively, the range proxy can be defined as

𝑅𝑃𝑡 =1

4 ln(2)(ln (

𝑃𝑡𝐻𝑖𝑔ℎ

𝑃𝑡𝐿𝑜𝑤 ))

2

, (2)

where 𝑃𝑡𝐻𝑖𝑔ℎ

and 𝑃𝑡𝐿𝑜𝑤 are the highest and lowest observed prices at time 𝑡, respectively. The ratio


𝑃𝑡𝐿𝑜𝑤⁄ illustrates the high-low range of an asset. In addition, the constant 1 (4 ln(2))⁄ is a scaling

factor, which represents the second moment of range of a Wiener process (Bannouh et al., 2009).

However, Eq. (2) does not incorporate the financial asset’s opening and closing prices on day 𝑡,

despite that they may improve the estimation of 24-hour volatility. Therefore, a more accurate range

proxy is available, assuming again that log returns are normally distributed, namely

𝑅𝑃𝑡∗ =

1

2(ln (


𝑃𝑡𝐿𝑜𝑤 ))

2

− (2 ln(2) − 1) (ln (𝑃𝑡

𝐶𝑙𝑜𝑠𝑒

𝑃𝑡𝑂𝑝𝑒𝑛))

2

. (3)

Note that 𝑃𝑡𝐶𝑙𝑜𝑠𝑒 and 𝑃𝑡

𝑂𝑝𝑒𝑛 are the observed closing and opening prices at time 𝑡, respectively. Alt-

hough the inclusion of opening and closing prices can improve the variance estimator, the gains are

not necessarily realised in practice because of the effects of market microstructure noise on asset

prices (Alizadeh et al., 2002; Martens & van Dijk, 2007). Yet, we will use these two RP measures in

this research as alternatives to the realised variance estimators.

As stated in Alizadeh et al. (2002) and Bannouh et al. (2009), the range-based approach has been

known as a volatility estimator for a long time. Parkinson (1980) has demonstrated that the extreme

value method, or high-low range, is noticeably more efficient as a variance estimator compared to the

squared return method. Other studies have exploited this result for development of volatility proxy

based on intraday high-low prices that appears to be far more efficient than the realised variance, see

Martens and van Dijk (2007) and Christensen and Podolskij (2007).


9

3.2. Forecasting models

Throughout the years, literature has propounded numerous forecast models that are to a certain

extent applicable to applied economic and financial research. As discussed in Christoffersen (2012),

the goal of a volatility-forecasting model is to predict future variance as precisely as possible to apply

necessary risk measures. In the past decades, many researchers have evaluated and analysed the fore-

casting performance of distinct volatility-forecasting models, ranging from simple moving average to

sophisticated stochastic models. An extensive review of 93 empirical studies can be found in Poon

and Granger (2003). We will focus on six GARCH-type models, namely the ARCH, GARCH, GJR-GARCH,

CGARCH, APARCH, and GARCHX models. We will discuss these forecast models in more detail in the

next subsubsections.

• 3.2.1. ARCH

In 1982, Engle introduced a new class of stochastic process called the Autoregressive Conditional

Heteroskedasticity (ARCH) process, which allows the conditional variance to be time varying as a

linear function of lagged errors, leaving the unconditional variance constant over time (Engle, 1982;

Bollerslev, 1986). ARCH was one of the first econometric models that provided a convenient way to

model conditional heteroskedasticity in variance. First, to model an ARCH process, let 𝜉𝑡 denote the

disturbance term, which depends on a stochastic component 𝑧𝑡 and a time-varying standard devia-

tion 𝜎𝑡 (Nelson, 1991). Mathematically, it can be written as

𝜉𝑡 = 𝜎𝑡𝑧𝑡 ,

where 𝑧𝑡 ~ i.i.d. 𝒩(0,1). By definition, 𝜉𝑡 is serially uncorrelated with mean zero and conditional var-

iance equal to 𝜎𝑡2. In mathematical terms, 𝜉𝑡|𝜓𝑡−1 ~ 𝒩(0, 𝜎𝑡

2) where 𝜓𝑡 denotes the set of all infor-

mation available at time 𝑡 (Engle, 1982). The conditional variance, 𝜎𝑡2, is modelled as follows

𝜎𝑡2 = 𝜔0 + ∑ 𝛼𝑗𝜉𝑡−𝑗

2

𝑞

𝑗=1

, (4)

where 𝜔0 > 0 and 𝛼𝑗 ≥ 0 ∀𝑗 ∈ {1,2, … , 𝑞}. In this model, 𝜎𝑡2 is expressed as a linear function of lagged

squared errors where recent disturbances matter more for current period’s conditional variance than

distant errors (Bollerslev et al., 1992). Hence, this model is referred as the 𝐴𝑅𝐶𝐻(𝑞) model.

The main strengths of the ARCH model are its simplicity and ability to incorporate well-known em-

pirical regularities such as heavy tails, volatility clustering, and mean reversion (Bera & Higgins,

1993). Eq. (4) captures the conditional and unconditional variances in 𝜎𝑡2 and 𝜔0, respectively. The

latter ensures that the model incorporates the mean reverting characteristic of the data. Despite its


10

novelty at that time, the model has several limitations as well. First, ARCH requires a high order of lag

variables to capture the dynamic behaviour of conditional variance in practice, see Bollerslev (1986),

Bollerslev et al. (1992), Bera and Higgins (1993), and Bollerslev et al. (1994). This is simply a compu-

tational burden for the modeller due to the estimations of many parameters that are subject to ine-

quality restrictions. Second, Engle’s model assumes the conditional variance to be a linear function of

lagged squared error terms that does not take into account the leverage effect (Nelson, 1991; Bera &

Higgins, 1993). In other words, 𝜎𝑡2 is affected only by the magnitude and not by the sign of 𝜉𝑡−𝑗. Ac-

cordingly, the ARCH model is not able to incorporate the asymmetric behaviour of volatility shocks.

• 3.2.2. GARCH

Not long after the introduction of Engle’s ARCH model, Bollerslev proposed a more generalised ver-

sion of this model in 1986, also called the Generalised Autoregressive Conditional Heteroskedasticity

(GARCH) model. Bollerslev’s GARCH model is an extension of Engle’s ARCH model that allows for

more flexible lag structure, i.e. the GARCH model requires less parameters to be estimated than the

ARCH model to forecast conditional variances (Bollerslev, 1986). Since the properties of the ARCH

and GARCH models are reasonably similar, 𝐺𝐴𝑅𝐶𝐻(𝑝, 𝑞) is defined as

𝜎𝑡2 = 𝜔0 + ∑ 𝜑𝑖𝜎𝑡−𝑖

2

𝑝

𝑖=1

+ ∑ 𝛼𝑗𝜉𝑡−𝑗2

𝑞

𝑗=1

, (5)

where 𝜔0, 𝜑𝑖 , and 𝛼𝑗 are nonnegative parameters. For a stable conditional variance process, we re-

quire the weights to be ∑ ∑ (𝜑𝑖 + 𝛼𝑗) < 1𝑞𝑗=1

𝑝𝑖=1 (Hull, 2012). In this case, the conditional variance, 𝜎𝑡

2,

is a linear function of past estimates of variance rates and past squared errors, where more recent

observations of both parameters get relatively higher weights compared to more distant observa-

tions. Interestingly, the conditional variance of 𝜉𝑡 behaves like an ARMA process (Enders, 2015). No-

tice that Eq. (4) is a special case of Eq. (5).

As noted by Nelson (1991), GARCH models elegantly capture the volatility clustering in asset re-

turns. Another important feature is that these models can be fitted to data that have excess kurtosis

(Franses & Ghijsels, 1999). Furthermore, these models recognise the mean-reverting dynamics of as-

set return volatility, which implies that the variance tends to converge to a long-run average variance

rate (Hull, 2012). More importantly, the GARCH model is often considered as a parsimonious model

since it uses less lag variables compared to the ARCH models in empirical applications (Enders, 2015).

The above features accentuate the key strengths of this econometric model.


11

Unfortunately, the GARCH model has its limitations as well. Similar to the ARCH model, the standard

GARCH model lacks the ability to incorporate the volatility asymmetry of positive and negative re-

turns (Nelson, 1991). Another limitation results from the nonnegativity constraints on the variance

equation parameters, see Eq. (5), which rule out any stochastic oscillatory behaviour in the 𝜎𝑡2 pro-

cess and can complicate the estimation of GARCH models. Lastly, the interpretation of the persistence

of volatility shocks may be questionable. If volatility shocks persist continuously, they may stir the

entire term structure of risk premia, and therefore are likely to affect significantly the investment in

tangible assets (Poterba & Summers, 1986).

• 3.2.3. GJR-GARCH

As stated previously, the standard GARCH model does not incorporate the asymmetric dynamics of

asset return shocks. Fortunately, a commonly used asymmetric GARCH-type model that captures this

leverage effect phenomenon is called the GJR-GARCH model, proposed by Glosten et al. (1993). Com-

pared to the normal GARCH model, GJR-GARCH is a richer model since it augments the GARCH model

with a dummy variable 𝐼𝑡−𝑗 that takes either a value of one or zero (Brailsford & Faff, 1996). Hence,

𝐼𝑡−𝑗 is mathematically defined as

𝐼𝑡−𝑗 = { 1 ⟸ 𝜉𝑡−𝑗 < 0

0 ⟸ 𝜉𝑡−𝑗 > 0.

As a result, we can specify the 𝐺𝐽𝑅-𝐺𝐴𝑅𝐶𝐻(𝑝, 𝑞) model as


2

𝑝

𝑖=1

+ ∑(𝛼𝑗 + 𝜆𝑗𝐼𝑡−𝑗)𝜉𝑡−𝑗2

𝑞

𝑗=1

, (6)

where 𝜔0, 𝜑𝑖 , 𝛼𝑗, and 𝜆𝑗 are nonnegative parameters. Besides, for a stable conditional variance pro-

cess, we set ∑ ∑ (𝜑𝑖 + 𝛼𝑗) < 1𝑞𝑗=1

𝑝𝑖=1 . When the indicator variable 𝐼𝑡−𝑗 > 0, it simply implies that past

negative return shocks have a larger impact on the future conditional standard deviations than past

positive return shocks (Andersen et al., 2006). Note that Eq. (6) turns into Eq. (5) when 𝐼𝑡−𝑗 = 0.

• 3.2.4. CGARCH

Another GARCH-type model that we will introduce is the component GARCH (henceforth CGARCH)

model. This model tries to decompose volatility into a short-run component and long-run component

(Engle & Lee, 1999; McMillan & Speight, 2004). Shocks of the long-run volatility component are highly

persistent, whereas shocks of the transitory component are relatively less persistent (Watanabe &

Harada, 2006). Normally, in a standard GARCH model, the conditional variance tends to mean revert


12

to its long-run variance component, 𝜔0, over time. In contrast, CGARCH allows mean reversion to a

time-varying trend (Engle & Lee, 1999). Thus, the 𝐶𝐺𝐴𝑅𝐶𝐻(𝑝, 𝑞) model is specified as

𝜎𝑡2 = �̃�𝑡 + ∑ 𝜑𝑖(𝜎𝑡−𝑖

2 − �̃�𝑡−𝑖)

𝑝

𝑖=1

+ ∑ 𝛼𝑗(𝜉𝑡−𝑗2 − �̃�𝑡−𝑗)

𝑞

𝑗=1

, (7)

where �̃�𝑡 and (𝜎𝑡−𝑖2 − �̃�𝑡−𝑖) are the time-varying trend and short-run component of 𝜎𝑡

2, respectively.

This time-varying long-run variance rate is formally defined as

�̃�𝑡 = ℎ0 + 𝑥�̃�𝑡−1 + 𝑦(𝜉𝑡−12 − 𝜎𝑡−1

2 ).

In this case, the forecasting error (𝜉𝑡−12 − 𝜎𝑡−1

2 ) drives the time-dependent movement of the trend

variable. Importantly, to achieve stationarity, we set (𝜑 + 𝛼)(1 − 𝑥) + 𝑥 < 1, which in turn requires

𝑥 < 1 and (𝜑 + 𝛼) < 1. As a result, the short-run component then converges to zero with powers of

𝜑 + 𝛼, while the long-run component converges to �̃�𝑡 with powers of 𝑥 (McMillan & Speight, 2004).

• 3.2.5. APARCH

Ding et al. (1993) proposed a general class of forecast model that includes Engle’s ARCH, Bollerslev’s

GARCH, and five other GARCH-type models as special cases. This polyvalent forecast model is also

called the Asymmetric Power Autoregressive Conditional Heteroskedasticity (APARCH) model. Thus,

the standard 𝐴𝑃𝐴𝑅𝐶𝐻(𝑝, 𝑞) model is formally defined as

𝜎𝑡𝛿 = 𝜔0 + ∑ 𝜑𝑖𝜎𝑡−𝑖

𝛿

𝑝

𝑖=1

+ ∑ 𝛼𝑗(|𝜉𝑡−𝑗| − 𝜆𝑗𝜉𝑡−𝑗)𝛿

𝑞

𝑗=1

(8)

where 𝜔0, 𝛿, 𝜑𝑖 , and 𝛼𝑗 are nonnegative parameters, meanwhile parameter 𝜆𝑗 ∈ (−1,1). Notice that 𝛿

is the Box-Cox transformation, or exponentiation, of the conditional variance and the asymmetric ab-

solute residuals (Ding et al., 1993; Laurent, 2004). This transformation is convenient for linearising

nonlinear models. Furthermore, parameter 𝜆𝑗 reflects the so-called leverage effect (Brooks et al.,

2000). When 𝜆𝑗 > 0, past negative shocks have a deeper effect on 𝜎𝑡 than past positive shocks. Mean-

while, 𝜆𝑗 < 0 implies that past positive shocks have a greater impact on 𝜎𝑡 than past negative shocks.

The key strength of the APARCH model is its ability to incorporate well-known empirical regulari-

ties, such as volatility clustering, leverage effect, and long-memory property, due to the flexible struc-

ture of the model which nests at least seven different ARCH-type models (Laurent, 2004). By changing

the underlying parameters of APARCH, we obtain the ARCH, GARCH, TS-GARCH, GJR-GARCH, TARCH,

NARCH, and Log-ARCH models (Ding et al., 1993). However, despite the model’s novelty and versatil-

ity, the APARCH model does not allow fractional integration of conditional variance. Consequently,


13

conditional variance shocks either dispel exponentially or persist indefinitely (Degiannakis, 2004).

Baillie et al. (1996) and Tse (1998) proposed extensions to the APARCH model to address this limita-

tion by expressing the conditional variance equation as a function of lagged shocks that decay at a

gradual hyperbolic rate.3

• 3.2.6. GARCHX

Throughout the years, many extensions to the classic GARCH model, ranging from nonlinear asym-

metric power modifications to multivariate class models, have been proposed in the finance litera-

ture. Just like the Capital Asset Pricing Model (Sharpe, 1964; Lintner, 1965; Mossin, 1966), Arbitrage

Pricing Theory (Ross, 1976), and Black-Scholes-Merton Model (Black & Scholes, 1973; Merton, 1973),

the GARCH model can be easily extended to a more complex form by adding external regressors into

the equation. In this paper, we will extend the 𝐺𝐴𝑅𝐶𝐻(𝑝, 𝑞) by adding a covariate linked to the Cboe

Volatility Index, or VIX, and call it the 𝐺𝐴𝑅𝐶𝐻𝑋(𝑝, 𝑞, 𝑟) model.4 The VIX Index reflects investors’ ex-

pectation of the volatility of the S&P 500 over the next month, which is implied by the current prices

of S&P 500 index options (Kambouroudis & McMillan, 2016).5 Thus, it can be seen as a forward-look-

ing volatility measure. Therefore, including VIX as an additional independent variable may signifi-

cantly improve the forecastability of a standard GARCH-type model (Christoffersen, 2012). Earlier

empirical studies by Fleming et al. (1995), Hol (2003), Blair et al. (2010), and Kambouroudis and

McMillan (2016) provided supportive evidence for VIX in providing accurate volatility forecasts.

Our 𝐺𝐴𝑅𝐶𝐻𝑋(𝑝, 𝑞, 𝑟) model is formally defined as


2

𝑝

𝑖=1

+ ∑ 𝛼𝑗𝜉𝑡−𝑗2

𝑞

𝑗=1

+ ∑ 𝜐𝑘휁𝑡−𝑘

𝑟

𝑘=1

, (9)

where variable 휁𝑡 =1

252(

𝑉𝐼𝑋𝑡

100)

2 is defined as a variance proxy for 𝑉𝐼𝑋𝑡 . The daily VIX variance proxy

is in accordance with Hao and Zhang’s (2013) approach. Further, 𝜐𝑘 is a weight of covariate 휁𝑡. Hence,

the conditional variance equation of GARCHX is now a linear function of lagged variances, squared

errors, and VIX variance proxy. Fortunately, we can estimate covariate 휁𝑡−1 with univariate maximum

likelihood estimation.

3 Baillie et al. (1996) and Tse (1998) introduced the FIGARCH and FIAPARCH, respectively, which are fraction-ally integrated extensions of the GARCH-type models. 4 Cboe: Chicago Board Options Exchange. 5 The Cboe VIX is often called “investor fear gauge” by financial news media. According to Cboe’s website, the VIX Index can be considered as the world’s premier barometer of investor sentiment and market volatility.


14

3.3. Model fitting & forecast evaluation

To assess the quality and predictive ability of forecast models, one should consider several tests for

both short- and medium-term predictions of conditional variance. Therefore, we will discuss more

about goodness-of-fit and forecast evaluation tests in the following subsubsections.

• 3.3.1. Information criteria tests

A typical goodness-of-fit test indicates whether a forecast model is statistically significant, but it

does not tell us how we should select between these models (Bozdogan, 1987; Kuha, 2004). Neither

does it penalise for potential overfitting when additional parameters are added to the model. Fortu-

nately, there are better equipped tests that avoid these types of issues such as the Akaike Information

Criterion (AIC) and Bayesian Information Criterion (BIC), see Burnham & Anderson (2004). These in-

sample fitting tests penalise for the number of model parameters to avoid possible overfitting issues.

AIC and BIC are used as relative quality measures for selecting the best-fitting model amongst others.

The forecast model with the smallest information criterion coefficient is preferred since it fits best

with the in-sample data. Thus, the AIC is formally defined as

𝐴𝐼𝐶 = 2𝑘 − 2 ln(𝜃), (10)

where 𝑘 is the number of model parameters and 𝜃 is the maximum value of the forecast model’s like-

lihood function (Bozdogan, 1987), meanwhile the BIC is defined as

𝐵𝐼𝐶 = ln(𝑛) 𝑘 − 2 ln(𝜃), (11)

where 𝑛 describes the quantity of sample information (Kuha, 2004).

• 3.3.2. Mean squared error

Similar to Brownlees et al. (2011), Bollerslev et al. (2016), and Quaedvlieg (2018), we will use the

mean squared error (henceforth MSE), or quadratic loss, measure to assess our conditional variance

forecasts. As discussed in Patton (2011b), mean squared error is a robust loss function. According to

Lopez (2001), MSE is a commonly used forecast evaluation tool for both in-sample and out-of-sample

forecasts. Simply put, MSE is the average squared difference between the realised conditional vari-

ance and the corresponding volatility forecast. Mathematically,

𝑀𝑆𝐸 =1

𝑛∑(𝑉𝐴�̂�𝑡+𝑖 − �̂�𝑡+ℎ|𝑡

2 )2

𝑛

𝑖=1

, (12)

where 𝑉𝐴�̂�𝑡+𝑖 and �̂�𝑡+ℎ|𝑡2 are the unbiased ex-post estimate of conditional variance and ℎ-step(s)

ahead forecast of conditional variance, respectively. Squaring the difference results in larger weights

for possible outliers. Dismally, this makes MSE sensitive to potential outliers. Eventually, the forecast


15

model with the lowest MSE loss coefficient is preferred. In our case, we scale the loss coefficients for

convenience sake by taking the square root of MSE, also known as root mean squared error (RMSE).

Unfortunately, basing forecast evaluation on the MSE approach results in two shortcomings. Firstly,

despite that 𝑉𝐴�̂�𝑡+𝑖 is an unbiased variance estimator, it is technically still an imprecise estimator

(Lopez, 2001). Secondly, MSE penalises conditional variance estimates that are distinct from the re-

alised conditional variance in a fully symmetrical way. It does not penalise for negative or zero esti-

mates of the conditional variance (Bollerslev et al., 1994). In practice, however, one may reckon con-

trastingly about negative and positive forecast errors of the same magnitude, as underestimation of

volatility may be costlier than overestimation by the same amount (Christoffersen, 2012).

• 3.3.3. Quasi-likelihood

To allow for asymmetric loss when evaluating volatility forecasts, one can use the quasi-likelihood

(henceforth QLIKE) loss function, which is another commonly used robust loss function in empirical

studies. Like mean squared error, forecast model with the lowest QLIKE loss is preferred. The QLIKE

is mathematically defined as

𝑄𝐿𝐼𝐾𝐸 =𝑉𝐴�̂�𝑡+𝑖

�̂�𝑡+ℎ|𝑡2 − log (

𝑉𝐴�̂�𝑡+𝑖

�̂�𝑡+ℎ|𝑡2 ) − 1. (13)

Note that this loss function depends on a relative forecast error, 𝑉𝐴�̂�𝑡+𝑖 �̂�𝑡+ℎ|𝑡2⁄ , whereas MSE depends

on an absolute forecast error (Brownlees et al., 2011). QLIKE will always penalise biased forecasts of

conditional variance, i.e. underestimation of volatility, more heavily than MSE (Christoffersen, 2012).

Here we first determine the model’s QLIKE losses for the entire out-of-sample period and then esti-

mate the model’s mean loss.

According to Brownlees et al. (2011), there are reasons why someone prefers QLIKE to MSE for

forecast comparison. First, since the QLIKE loss function depends on a relative volatility forecast er-

ror, the loss series is i.i.d. under the null, assuming that the forecast model is specified properly. MSE

on the other hand, which depends entirely on 𝑉𝐴�̂�𝑡+𝑖 − �̂�𝑡+ℎ|𝑡2 , will have a variance that is propor-

tional to the square of the variance of asset returns (Patton, 2011b), and thus consists of high levels

of serial dependence even under the null. Second, MSE has a bias that is proportional to the square of

the true variance, whereas QLIKE bias is independent of the volatility level (Brownlees et al., 2011).

Amidst severe market turmoil, sizable MSE loss coefficients will be a repercussion of high volatility

regime without necessarily corresponding to the decay of a model’s predictive ability. QLIKE avoids

this vagueness, making it simpler to study forecast errors across different volatility regimes.


16

Nevertheless, both MSE and QLIKE loss functions do not tell anything about the statistical signifi-

cance of forecast performances, making predictive comparisons incomplete (Diebold, 2015). The dif-

ference between loss coefficients may not be significantly different from zero (Choudhry & Wu, 2008).

Therefore, to draw any meaningful inferences about the predictive ability of forecast models, we also

need suitable significance tests.

• 3.3.4. Diebold-Mariano test

The Diebold-Mariano (DM) test is a commonly used significance test for pairwise comparisons of

competing forecasts at different horizons (Diebold & Mariano, 1995). Compared to the abovemen-

tioned loss functions, the DM test allows one to assess the statistical significance of apparent predic-

tive superiority (Diebold, 2015). This method relies on assumptions made directly on the difference

between two loss functions, also called loss differential. The DM test requires the loss differential to

be covariance stationary, but it may not be strictly necessary in some cases (Diebold, 2015). For any

loss function, the loss differential, 𝑑𝑖𝑗,𝑡, can be defined as

𝑑𝑖𝑗,𝑡 = 𝐿𝑖,𝑡 − 𝐿𝑗,𝑡 , (14)

with 𝐿𝑖,𝑡 and 𝐿𝑗,𝑡 as loss functions of forecast models 𝑖 and 𝑗, respectively, at time 𝑡. In case when the

disparity in the accuracy of two competing models is zero, then 𝔼(𝐿𝑖,𝑡) = 𝔼(𝐿𝑗,𝑡), or 𝔼(𝑑𝑖𝑗,𝑡) = 0. This

corresponds with the equal predictive accuracy null hypothesis, which states that the population

mean of a loss differential series is equal to zero (Diebold & Mariano, 1995). To test this null hypoth-

esis, a simple asymptotic 𝑧-test can be used. Therefore, the DM test statistic can be obtained through

𝐷𝑀 =�̅�𝑖𝑗

�̂��̅�𝑖𝑗

, (15)

where �̅�𝑖𝑗 =1

𝑇∑ 𝑑𝑖𝑗,𝑡

𝑇𝑡=1 , or sample mean loss differential, and �̂��̅�𝑖𝑗

is a consistent estimator of the

standard deviation of �̅�𝑖𝑗 . Under the null hypothesis, the DM test statistic has an asymptotic standard

normal distribution (Harvey et al., 1997). Importantly, the DM statistic can be obtained by regression

of the loss differential on a constant, using Newey-West standard errors (Diebold, 2015).6

However, just like other statistical metrics, the DM test is subject to flaws as well. Firstly, the initial

DM test was found to be oversized for moderate numbers of sample observations (Harvey et al.,

1997). Yet, the DM test is found considerably more versatile than other tests of equal forecast accu-

racy. When forecast errors are not normally distributed, the Morgan-Granger-Newbold test (1977)

6 Newey-West standard errors are also called heteroskedasticity and autocorrelation consistent (HAC) robust standard errors (Newey & West, 1994).


17

and Meese-Rogoff test (1988) for equal MSE are severely missized in both large and small samples.

The DM test, on the other hand, maintains approximately correct size for all but moderate-sized sam-

ples (Mariano, 2002). Secondly, reporting results of pairwise comparisons becomes rather laborious

when the set of models increases, since one must perform 𝑛(𝑛 − 1) 2⁄ tests (Hansen et al., 2003).

• 3.3.5. Model Confidence Set

Contrary to the widely used Diebold-Mariano test, the Model Confidence Set (MCS), introduced by

Hansen et al. (2003), allows for comparison of multiple forecast models at once. This model selection

method is an innovative way to deal with the issue of selecting the “best” forecast model(s) using out-

of-sample evaluation under a specified loss function. As defined in Hansen et al. (2011), an MCS, or

ℳ∗, is a subset of a collection of candidate models, ℳ0, which consists of superior forecast models

for a given significance level. The set of superior forecast models is formally defined as

ℳ∗ = {𝑖 ∈ ℳ0: 𝜇𝑖𝑗 ≤ 0, ∀ 𝑗 ∈ ℳ0}, (16)

where 𝜇𝑖𝑗 = 𝔼(𝑑𝑖𝑗,𝑡) is finite and does not depend on 𝑡 for all 𝑖, 𝑗 ∈ ℳ0. Note that 𝑑𝑖𝑗,𝑡 is the loss dif-

ferential. The aim of this method is to determine the set of superior models, which can be done via a

sequence of significance tests where models that are found to be significantly inferior to other models

of ℳ0 are eliminated (Hansen et al., 2011). Hence, MCS can be viewed as a sequential DM test

(Quaedvlieg, 2018), or as a confidence interval of a parameter (Samuels & Sekkel, 2013). It is appeal-

ing to use a set of forecast models rather than an individual model, since there is no generic model

that will consistently outperform other models in all conceivable scenarios.

This procedure is constructed from an equivalence test, 𝛿ℳ , and an elimination rule, 𝑒ℳ , that are

assumed to have certain properties (Hansen et al., 2011). The former is used to test whether the null

𝐻0,ℳ: 𝜇𝑖𝑗 = 0, ∀ 𝑖, 𝑗 ∈ ℳ with ℳ ⊆ ℳ0, while the latter identifies an inferior model from ℳ and re-

moves it when 𝐻0 is rejected. Eventually, the surviving models end up in ℳ̂∗ after sequentially trim-

ming ℳ0 whilst holding the significance level, 𝛼, fixed at each procedural step (Hansen et al., 2003).

The algorithm for constructing ℳ̂1−𝛼∗ is as follows

Step I. Initiate the procedure by setting ℳ = ℳ0.

Step II. Test null hypothesis by using 𝛿ℳ at level 𝛼.

a. If 𝐻0,ℳ is not rejected, define ℳ̂1−𝛼∗ = ℳ.

b. If 𝐻𝑎,ℳ: 𝜇𝑖𝑗 ≠ 0 is accepted, then use 𝑒ℳ to remove forecast model 𝑖 from ℳ, and

reiterate Step II of this procedure.


18

To test the null hypothesis, we need to define 𝛿ℳ , which is simply based on the well-known Diebold-

Mariano test, see Eq. (15). In this case, �̂��̅�𝑖𝑗

2 is a bootstrapped estimate of the variance of the sample

mean loss differential that can be computed through a block bootstrap method of 𝐵 resamples with a

certain block length of ℒ (Bernardi & Catania, 2018). Moreover, 𝑒ℳ is defined as

𝑒𝑚𝑎𝑥,ℳ = arg max𝑖∈ℳ

�̅�𝑖

�̂��̅�𝑖

. (17)

Interestingly, this procedure yields 𝑝-values for all forecast models under consideration (Hansen et

al., 2003). These MCS 𝑝-values are useful since then it is clear which forecast models are included in

ℳ̂1−𝛼∗ for any significance level. The 𝑝-value for 𝑒ℳ𝑗

∈ ℳ0, or forecast model 𝑗, is formally defined by

�̂�𝑒ℳ𝑗= max

𝑖≤𝑗 𝑝𝐻0,ℳ𝑖

, where 𝑝𝐻0,ℳ𝑖 stands for the 𝑝-value that is associated with 𝐻0,ℳ𝑖

. Forecast models

with relatively modest 𝑝-values are not probable to be contained in ℳ̂1−𝛼∗ , because �̂�𝑖 ≥ 𝛼 for any 𝑖 ∈

ℳ0 in order to be part of ℳ̂1−𝛼∗ , see Hansen et al. (2011).

One of the benefits of the Model Confidence Set is that it acknowledges the limitations of the data

(Hansen et al., 2003, 2011). In other words, informative data will lead to ℳ∗ that holds only the best

model, whereas less informative data make it hard to differentiate between forecast models. Hence,

this method is dissimilar to other model selection criteria that select a single model and disregard the

surprisal of the underlying data. Another benefit is that this procedure makes it feasible to make state-

ments regarding the significance that are valid in the conventional sense, a characteristic which is not

satisfied by the frequently used approach of listing 𝑝-values from various pairwise comparison tests

(Hansen et al., 2011). Furthermore, the MCS approach allows for the possibility that ℳ∗ contains

more than one superior model. Lastly, ℳ∗ does not discard a forecast model unless it is found to be

significantly inferior in comparison with other models (Hansen et al., 2003).

3.4. Forecasting procedure

Before we start using these GARCH-type models, the forecasting procedure needs to be outlined

first. We split the data sample into two separate periods, namely in-sample and out-of-sample peri-

ods. The in-sample period starts from January 5, 2000, to August 6, 2004, whereas the out-of-sample

starts from August 9, 2004, to March 27, 2018. The former results in 1,000 daily observations that

covers periods of relatively large and small movements in volatility, meanwhile the latter has 3,078

out-of-sample observations. Hence, there are 4,078 daily observations in total (see Table 1). Model

predictions that are based entirely on in-sample data are susceptible to potential overfitting, as in-

sample forecast evaluations tend to spuriously indicate the existence of predictability when there is


19

none (Inoue & Kilian, 2005). In other words, in-sample forecasts draw an unduly optimistic picture

of a model’s predictive power. Including out-of-sample observations will improve the predictions of

a forecast model, as the power of forecast evaluation tests is strongest with long out-of-sample peri-

ods (Hansen & Timmermann, 2012). Therefore, to avoid potential overfitting issues, we will split the

sample into two separate periods.

Table 1. – Data sample

Data sample Begin date End date Observations

Full sample 01/05/2000 03/27/2018 4,078

In-sample period 01/05/2000 08/06/2004 1,000

Out-of-sample period 08/09/2004 03/27/2018 3,078

Table 1. – Brief summary of the different sample periods used in this paper.

Next, we will specify the following volatility-forecasting models in R by using the “rugarch” package:

ARCH(1), GARCH(1,1), GJR-GARCH(1,1), CGARCH(1,1), APARCH(1,1), and GARCHX(1,1,1).7 Since the

order structure of these conditional variance equations offer numerous combinations, we focus only

on parsimonious forecast models because of their computational efficiency and simplicity. Notice that

this paper tries to find out what makes a specific GARCH model so good in volatility forecasting rather

than discovering a superior model that has the best specifications by chance. Therefore, we will use

only one lag order for all forecast models.

Thereafter, we will conduct rolling window direct forecasts with daily re-estimations of model pa-

rameters under Gaussian distribution to predict ℎ-step(s) ahead out-of-sample conditional variances,

or �̂�𝑡+ℎ|𝑡2 . According to Christoffersen (2012), a general rule of thumb is to use a forecasting origin of

at least 1,000 observations. Therefore, for the re-estimations of model parameters, we will use a roll-

ing window of 1,000 observations that moves one-step forward each time. We are interested to fore-

cast �̂�𝑡+ℎ|𝑡2 at different horizons to evaluate the short- and medium-term predictive ability of several

GARCH models. The horizons used in this paper are one-day, one-week, one-month, and one-quarter.8

Lastly, we will evaluate �̂�𝑡+ℎ|𝑡2 series by using RMSE and QLIKE, and eventually assess the significance

of these evaluation tests by using DM tests and MCS.

7 See Ghalanos (2019) for documentation of this R package. 8 Here we assume that a week has five trading days, whereas a month and a quarter have, on average, 21 and 63 trading days, respectively, see Christoffersen (2012).


20

§4. Data

In this paper, we will use major equity indices from four different countries, of which two from

developed markets and two from emerging markets. Therefore, we will use the S&P 500, HSI, IPC, and

KOSPI indices in our research, respectively. Time series data such as index prices and realised vola-

tility estimations are retrieved from Oxford-Man Institute’s realised library. The database illustrates

a variety of realised volatility estimations that are based on different sampling methods. This paper

uses the 5-minute and 10-minute subsampled realised variances as proxies. Furthermore, to deter-

mine the range-based variance proxies, the high and low prices of these four equity indices are re-

trieved from Bloomberg. The daily log returns of equity indices are calculated as 𝑟𝑡 = ln(𝑃𝑡 𝑃𝑡−1⁄ ), see

Hull (2012). Lastly, daily prices of the Cboe Volatility Index, or VIX, are retrieved from Bloomberg as

well. The sample period of this novel dataset starts from January 5, 2000, to March 27, 2018, which

yields 4,078 daily observations in total, as Oxford-Man Institute’s database starts from January 2000

onwards. Importantly, due to market holidays and possible mistakes in data entry, all rows that con-

tain at least one missing value are left out from the sample. In other words, if a certain equity index

has a missing value at time 𝑡 + 𝑘, then the corresponding row including the non-missing values of

other equity indices will be omitted.

Table 2 illustrates the general information of the equity indices that are used in this paper. The New

York Stock Exchange (NYSE) and Stock Exchange of Hong Kong (SEHK) are considered developed

markets, whereas Bolsa Mexicana de Valores (BMV) and Korea Exchange (KRX) are labelled as emerg-

ing markets (MSCI, 2018).

Table 2. – List of stock indices

Country Stock exchange Major stock index Constituents

Hong Kong Stock Exchange of Hong Kong Hang Seng Index (HSI) 50

Mexico Bolsa Mexicana de Valores Índice de Precios y Cotizaciones (IPC) 35

South Korea Korea Exchange Korea Composite Stock Price Index (KOSPI) 751

United States New York Stock Exchange Standard & Poor’s 500 (S&P 500) 500

Table 2. – The first, second, third, and fourth columns display the country, stock exchange, stock index, and num-

ber of constituents, respectively. The New York Stock Exchange and Stock Exchange of Hong Kong are considered

developed markets, while the remaining two are labelled as emerging markets according to MSCI (2018).

In the next subsection, descriptive statistics of the sample data will be explained in more detail.

Additional descriptions of certain time series and their corresponding tables and graphs can be found

in the Appendix sections.


21

4.1. Descriptive statistics

Table 3 displays summary statistics of the daily logarithmic returns of these stock indices. We ob-

serve that return distributions of these stock indices tend to follow a leptokurtic distribution since

the kurtosis estimates are all greater than three. Further, the average logarithmic returns and stand-

ard deviations of these indices tend to be close to zero and one, respectively. In addition, the Jarque-

Bera test statistics show that normality assumption of the sample data is rejected at a 1% significance

level in every case, which implies that the sample stock returns are likely to follow a non-normal dis-

tribution.9 In addition, Figure 1 illustrates the kernel density plots of return distributions of these

equity indices. These kernel density plots confirm the leptokurtic shape of the return distributions,

which is not surprising because stock returns tend to have fat tails (see Section 2.2.1). Note that the

return distribution of the KOSPI (see Panel D) is slightly skewed to the left compared to other indices.

Table 3. – Summary statistics of stock index returns

Indices Mean (%) Std. dev. (%) Min. (%) Max. (%) Skewness Kurtosis Jarque-Bera

S&P 500 0.015 1.280 -9.930 10.642 -0.279 11.404 12,054

HSI 0.014 1.584 -14.695 13.407 -0.392 12.625 15,846

IPC 0.048 1.380 -12.126 10.441 -0.176 10.251 8,955

KOSPI 0.021 1.606 -16.935 11.245 -0.739 11.613 12,976

Table 3. – This table illustrates the summary statistics of the daily logarithmic returns of various stock indices. The sample

period of these stock index returns starts from January 5, 2000, to March 27, 2018, which in turn gives 4,078 observations of

daily trading returns. Note that the above Jarque-Bera test statistics reject the null hypothesis, which states that the sample

data is normally distributed, in each case at a significance level of 1%.

The correlations of the four equity indices returns are demonstrated in Table 4. From the table, we

observe that returns of equity indices from the same continent tend to be stronger correlated with

each other than equity indices from other continents. For instance, S&P 500 has a positive correlation

coefficient of 0.69 with IPC, while the correlations with HSI and KOSPI are significantly weaker. One

could suggest using dynamic conditional correlations but that is beyond the scope of this paper.

Table 4. – Correlations of stock indices returns

Indices HSI IPC KOSPI S&P 500

HSI 1.00 IPC 0.33 1.00 KOSPI 0.63 0.30 1.00 S&P 500 0.27 0.69 0.22 1.00

Table 4. – This matrix displays the correlation coefficients of equity indices returns.

The data sample period starts from January 5, 2000, to March 27, 2018, which contains

4,078 observations in total.

9 Further explanation about the Jarque-Bera normality test is provided in Appendix Section A.1.


22

Figure 1. – Kernel density plots of stock index returns

Figure 1. – This graph illustrates the return distribution of the S&P 500, IPC, HSI, and KOSPI indices. In each panel, the navy-

blue curve illustrates the standard normal distribution meanwhile the red-coloured dashes display the kernel density. The

sample period of these stock returns starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.

Furthermore, Table 5 illustrates the summary statistics of the realised and range-based volatility

proxies. The distributions of both RV and RP tend to show a non-normal shape as the skewness and

kurtosis figures are significantly large. This also confirmed by the significantly large Jarque-Bera test

statistics. Notice that the standard deviation of 5-minute and 10-minute subsampled realised volatil-

ity series of the S&P 500 is substantially larger compared to other indices in Panels A and B, mean-

while in Panels C and D the two range-based variance proxies of the KOSPI seem to have the largest

standard deviations. Besides, it seems that the outliers of range-based variance proxies of these eq-

uity indices tend to be relatively more dispersed than the realised variance ones.


23

Table 5. – Summary statistics of volatility proxies

Indices Mean (%) Std. dev. (%) Min. (%) Max. (%) Skewness Kurtosis Jarque-Bera

Panel A: RV (5-min) S&P 500 1.118 2.583 0.012 77.477 11.671 239.684 9,611,194 HSI 1.019 1.775 0.048 43.730 10.280 172.901 4,976,700 IPC 0.871 1.890 0.044 52.248 12.491 249.963 10,469,385 KOSPI 1.271 2.256 0.057 59.437 9.297 161.995 4,354,140

Panel B: RV (10-min) S&P 500 1.139 2.616 0.013 77.841 11.228 228.166 8,700,413 HSI 1.018 1.832 0.038 55.101 12.163 259.519 11,281,405 IPC 1.016 2.093 0.041 45.485 9.757 144.092 3,447,227 KOSPI 1.278 2.266 0.041 62.899 9.674 181.920 5,503,042

Panel C: RP (Eq. 2) S&P 500 1.030 2.490 0.008 42.884 9.164 115.932 2,224,133 HSI 1.010 2.679 0.029 112.325 22.467 799.842 108,232,879 IPC 1.181 2.220 0.021 43.812 7.593 95.862 1,504,436 KOSPI 1.275 2.880 0 90.512 13.316 308.202 15,947,958

Panel D: RP (Eq. 3) S&P 500 0.912 2.272 0.008 59.205 11.526 199.766 6,668,932 HSI 0.964 2.440 0.030 103.590 22.301 810.743 111,199,939 IPC 1.014 1.841 0.023 42.914 8.171 113.806 2,131,610 KOSPI 1.235 2.891 -0.432 102.172 15.047 410.165 28,323,237

Table 5. – This table illustrates the summary statistics of different volatility proxies, namely the realised and range-based vari-

ances. The sample period starts from January 5, 2000, to March 27, 2018, which results in 4,078 daily observations for all proxies.

Note that the Jarque-Bera test statistics reject the null hypothesis in every case at a significance level of 1%.

Since the aim of this paper is to find out the properties of different forecast models, we are highly

interested in the underlying volatility of these equity indices. For the sake of brevity, figures of S&P

500 will be demonstrated in this subsection. Figure 2 illustrates the annualised volatility of the S&P

500 index on a daily basis over the period January 5, 2000, to March 27, 2018. The computations of

annualised volatilities are done similarly as in Patton (2011a) using 𝜎𝑡𝐴 = √252 × 𝑉𝐴�̂�𝑡, where 𝑉𝐴�̂�𝑡

is the variance proxy at time 𝑡. The figure illustrates volatility spikes that are scattered over the sam-

ple period. However, the magnitude of these volatility spikes varies among the four volatility proxies

at certain moments. For instance, Panels A and B tend to display larger volatility spikes during the

financial crisis of 2008 compared to Panels C and D. Additionally, the panels illustrate the clustering

of volatility in certain periods as well. It seems that sizable changes in variance tend to be followed by

relatively large changes, while small changes in variance tend to be followed by relatively small

changes.10 Similar results for volatility clustering can be found for other equity indices as well. The

10 Mandelbrot (1963) used equivalent words to outline the observation of volatility clustering of price variation. His exact words were: “…large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes…” (p. 418).


24

corresponding line plots of these equity indices are reported in Appendix Section C (see Figure C1,

Figure C2, and Figure C3).

Figure 2. – Annualised volatility proxies of S&P 500 index

Figure 2. – This figure displays volatility estimates of the S&P 500 index based on four different proxies, namely the 5-minute

and 10-minute RV, and the two range-based variances based on Eq. (2) and Eq. (3). The volatility estimates are annualised,

using the following formula from Patton (2011a): 𝜎𝑡𝐴 = √252 × 𝑉𝐴�̂�𝑡. Furthermore, the vertical axes in all panels are scaled up

by 1/100. The sample period starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.

In addition, Figure 3 illustrates the autocorrelation plots, or correlograms, of the four variance es-

timators of the S&P 500. We observe a significant decay in the autocorrelation coefficients of the re-

alised variance and range-based variance proxies after dozens of lags. After a while, the autocorrela-

tion of these volatility proxies tends to converge to zero in all four panels. Similar patterns can be

found for the other equity indices as well; see Figure C4, Figure C5, and Figure C6 in the Appendix.

These correlograms suggest that asset return volatility of the S&P 500, HSI, IPC, and KOSPI indices

tends to be persistent, given the relatively high autocorrelation values in the panels below. Hence, the

existence of long-memory property for these equity indices is probable.


25

Figure 3. – Autocorrelations of S&P 500 volatility proxies

Figure 3. – In this figure, correlograms of the four different volatility proxies of the S&P 500 are demonstrated in the above

panels. A window of 500 lags is used in all four panels. Furthermore, the grey dashes represent the Bartlett’s formula for 𝑀𝐴(𝑞)

95% confidence bands, i.e. 5% significance limits.

We find the partial autocorrelation plots of the four different variance estimators of the S&P 500

index in Figure 4. Again, we observe clear signs of serial correlation from the graph panels, but com-

pared to the previous figure the partial autocorrelation function illustrates a relatively fast decaying

pattern in these four panels. The serial correlation coefficients of these volatility proxy series con-

verge to zero just after several lags and remain relatively stable thereafter. Similar patterns can be

found for other equity indices as well; see Figure C7, Figure C8, and Figure C9 in the Appendix.

Compared to the previous correlograms, the partial autocorrelation plots suggest that persistence in

asset return volatility does not last very long.


26

Figure 4. – Partial autocorrelations of S&P 500 volatility proxies

Figure 4. – In this figure, partial autocorrelations of the four different volatility proxies of the S&P 500 are demonstrated in the

above panels. A window of 40 lags is used in all four panels. Furthermore, the grey dashes represent the 95% confidence bands,

which can be calculated as 𝑠𝑒 = 1 √𝑛⁄ .

Moreover, to check quantitatively the occurrence of serial correlation in time series, we can use the

Ljung-Box Q-test.11 The results are demonstrated in Table A3 in the Appendix. In short, based on

these test results, there is sufficient statistical evidence to conclude that these variance estimators

exhibit serial correlation. As an alternative to the Ljung-Box Q-test, this paper also uses Engle’s ARCH

test to check for serial dependence in the squared residual series.12 In addition, Table A4 illustrates

the results of Engle’s ARCH test. We find sufficient statistical evidence to conclude that the squared

errors of the equity indices returns tend to be serially correlated.

11 Further description regarding the Ljung-Box test is provided in Appendix Section A.3. 12 Engle’s ARCH test is described in more detail in Appendix Section A.4.


27

Figure 6, on the next page, illustrates the index price level, logarithmic returns, and squared returns

of the S&P 500, which are displayed in Panels A, B, and C, respectively. Interestingly, it seems that a

downward trend in the index price causes significant spikes in the daily returns, which tend to concur

with the considerable jumps in the volatility series, or squared return in this case. Notice that squared

return is a noisy proxy measure of volatility. Yet, squared return does stress the dispersion of asset

returns in a convenient way. This graphical finding seems to coincide with the notion of leverage ef-

fect, which is explained earlier in Section 2.2.4. Similar results can be found for the other equity in-

dices as well; see Figure C10, Figure C11, and Figure C12 in the Appendix. Additionally, Figure 5

displays the historical price pattern of the VIX. It seems that the VIX Index has the tendency to follow

the opposite trend relative to the S&P 500. The VIX can be used as a variance proxy as well, which is

shown in Figure C13 from the Appendix. We observe that the VIX variance proxy closely tracks the

realised variance and range-based variance proxies.

Figure 5. – Historical behaviour of the VIX Index

Figure 5. – Cboe Volatility Index, or VIX. The sample period starts from January 5, 2000, to March 27, 2018, which

contains 4,078 observations. This results in a sample mean price of $19.83 with minimum and maximum prices of

$9.14 and $80.86, respectively, and a standard deviation of $8.80.


28

Figure 6. – S&P 500 index

Figure 6. – This figure illustrates the historical data of the S&P 500 index price level, log returns, and squared returns. These are displayed in

Panels A, B, and C, respectively. The sample period starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.


29

§5. Empirical results

5.1. In-sample fitting results

Table 6 illustrates the in-sample fitting results of the GARCH-type models. Based on the goodness-

of-fit tests, the overall results seem to be mixed as there is no single model that fits the equity indices

best in all cases. We observe that GARCHX(1,1,1) is the best-fitting model in our subset of volatility-

forecasting models for S&P 500, meanwhile the APARCH(1,1) model fits the in-sample data of IPC and

KOSPI best. For HSI, the APARCH(1,1) model is preferred under the log-likelihood criteria but the

information criteria tests suggest that GJR-GARCH(1,1) is a better fitting model.

Table 6. – Results of in-sample goodness-of-fit tests

Forecast model Log-likelihood AIC BIC

Panel A: S&P 500 ARCH(1) -1,768.47 3.543 3.558 GARCH(1,1) 2,918.10 -5.828 -5.809 GJR-GARCH(1,1) 2,930.49 -5.851 -5.826 CGARCH(1,1) 2,917.74 -5.824 -5.794 APARCH(1,1) 2,936.22 -5.860 -5.831 GARCHX(1,1,1) 2,951.96*** -5.894*** -5.869***

Panel B: IPC ARCH(1) 2,687.13 -5.368 -5.354 GARCH(1,1) 2,757.79 -5.508 -5.488 GJR-GARCH(1,1) 2,768.45 -5.527 -5.502 CGARCH(1,1) 2,772.69 -5.533 -5.504 APARCH(1,1) 2,776.79*** -5.542*** -5.512*** GARCHX(1,1,1) 2,759.08 -5.508 -5.484

Panel C: HSI ARCH(1) 2,726.03 -5.446 -5.431 GARCH(1,1) 2,764.30 -5.521 -5.501 GJR-GARCH(1,1) 2,786.66 -5.563*** -5.539*** CGARCH(1,1) 2,773.50 -5.535 -5.506 APARCH(1,1) 2,786.91*** -5.562 -5.532 GARCHX(1,1,1) 2,765.63 -5.521 -5.497

Panel D: KOSPI ARCH(1) 2,362.99 -4.720 -4.705 GARCH(1,1) 2,387.86 -4.768 -4.748 GJR-GARCH(1,1) 2,401.64 -4.793 -4.769 CGARCH(1,1) 2,400.67 -4.789 -4.760 APARCH(1,1) 2,408.77*** -4.806*** -4.776*** GARCHX(1,1,1) 2,403.11 -4.796 -4.772

Table 6. – This table illustrates the results of in-sample goodness-of-fit tests for the above-

mentioned volatility-forecasting models. The in-sample period starts from January 5, 2000,

to August 6, 2004, resulting in 1,000 observations. *** indicates the most preferred fore-

casting model in terms of in-sample fit under the log-likelihood, AIC, and BIC criteria.

Furthermore, Table B1 from the Appendix illustrates the in-sample estimations of model param-

eters of all forecasting models. The results show mixed results for some GARCH-type models. For in-

stance, we observe that the APARCH(1,1) model has several statistically significant parameters for all


30

equity indices, meanwhile the GARCHX(1,1,1) model has in some cases only a few statistically signif-

icant parameters. The former result can possibly be explained by the model’s flexibility that allows

for many forms. It is important to mention that these in-sample estimations do not say anything about

predictive ability of our GARCH-type models.

5.2. Forecast evaluation

This subsection discusses the direct forecast results of the GARCH-type models at multiple horizons.

Again, only tables and figures related to the results of S&P 500 will be presented here for the sake of

brevity. The rest can be found in the Appendix sections.

• 5.2.1. Results of loss functions

Table 7 illustrates the forecast loss coefficients of the GARCH-type models of S&P 500 at multiple

horizons. Similar to Brownlees et al. (2011), when ℎ increases, the loss coefficients tend to show

larger discrepancies in forecast errors due to the uncertainty over longer horizons. From all the fore-

casting models, we clearly observe that the GARCHX(1,1,1) model has the lowest loss coefficients in

all cases, suggesting that this model predicts relatively more accurate forecasts of both short- and

medium-term conditional variances for the S&P 500 index than other models in this sample. Reason

why the GARCHX model performs relatively well here has probably to do with the inclusion of the VIX

variance proxy, since it tracks the RV and RP proxies relatively well (see Figure C13 in the Appendix),

resulting in sharper volatility forecasts and thus smaller discrepancies in forecast errors. In contrast,

the forecast results of the ARCH(1) model are relatively weak compared to other models in all cases.

Yet, it is no surprise that the ARCH(1) model underperforms here, since other forecast models are

extensions of the ARCH model, see Section 3.2.

Moreover, APARCH(1,1) seems to be a good alternative, as in second best model, for GARCHX(1,1,1)

when it comes to one-step ahead forecasts, as both RMSE and QLIKE coefficients of APARCH(1,1) are

second lowest for all variance proxies in this sample. However, for ℎ > 1, other forecasting models

are preferred as alternative for the GARCHX(1,1,1) model, due to their relatively small loss coeffi-

cients. For instance, the CGARCH(1,1) model has the second lowest RMSE coefficients for all variance

proxies. Meanwhile, QLIKE ranks the ordinary GARCH(1,1) model as second best under both the RV

and RP criteria for ℎ = 21 and ℎ = 63. For one-week ahead forecasts, QLIKE ranks APARCH(1,1) and

GJR-GARCH(1,1) as second best based on RV proxies and RP proxies, respectively.

Figure 7 and Figure 8 (see pages 34 and 35) illustrate the one-step ahead conditional variance

forecasts of the S&P 500. Notice that the red and blue lines display the in-sample forecasts and out-

of-sample forecasts, respectively. The ARCH(1) forecasts are substantial in size compared to other


31

GARCH models. In contrast to other GARCH forecasts, the ARCH(1) series seems to be less refined as

well, which is not surprising since the ARCH model is only a linear function of past squared errors. In

addition, we observe that the GARCHX(1,1,1) forecasts tend to be relatively small compared to the

other GARCH models in this sample.

Table 7. – Forecast loss coefficients of S&P 500

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑

Forecast model RMSE QLIKE RMSE QLIKE RMSE QLIKE RMSE QLIKE

Panel A: RV (5-min) ARCH(1) 6.280 1.139 6.617 1.139 6.807 1.258 6.963 1.481 GARCH(1,1) 2.238 0.352 2.568 0.437 3.143 0.601 3.648 0.754 GJR-GARCH(1,1) 2.062 0.319 2.534 0.421 3.181 0.623 3.728 0.770 CGARCH(1,1) 2.170 0.334 2.477 0.429 3.014 0.635 3.495 0.775 APARCH(1,1) 2.045 0.315 2.498 0.416 3.162 0.677 3.708 0.772 GARCHX(1,1,1) 1.915*** 0.267*** 2.281*** 0.375*** 2.705*** 0.567*** 3.047*** 0.720***

Panel B: RV (10-min) ARCH(1) 6.262 1.151 6.616 1.156 6.798 1.279 6.958 1.475 GARCH(1,1) 2.238 0.371 2.547 0.452 3.133 0.613 3.638 0.761 GJR-GARCH(1,1) 2.069 0.338 2.508 0.433 3.175 0.638 3.720 0.775 CGARCH(1,1) 2.166 0.353 2.453 0.443 3.002 0.648 3.485 0.780 APARCH(1,1) 2.045 0.334 2.471 0.431 3.156 0.694 3.700 0.776 GARCHX(1,1,1) 1.907*** 0.286*** 2.265*** 0.387*** 2.700*** 0.578*** 3.031*** 0.726***

Panel C: RP (Eq. 2) ARCH(1) 6.529 1.270 6.596 1.319 6.788 1.520 6.925 1.593 GARCH(1,1) 2.313 0.539 2.557 0.601 3.053 0.749 3.585 0.877 GJR-GARCH(1,1) 2.177 0.505 2.529 0.577 3.103 0.769 3.673 0.886 CGARCH(1,1) 2.227 0.522 2.436 0.589 2.914 0.780 3.422 0.888 APARCH(1,1) 2.148 0.502 2.486 0.578 3.083 0.833 3.653 0.896 GARCHX(1,1,1) 1.864*** 0.438*** 2.171*** 0.528*** 2.566*** 0.701*** 2.926*** 0.832***

Panel D: RP (Eq. 3) ARCH(1) 6.438 1.270 6.630 1.286 6.758 1.379 6.907 1.560 GARCH(1,1) 2.380 0.535 2.645 0.587 3.071 0.723 3.521 0.827 GJR-GARCH(1,1) 2.274 0.502 2.627 0.564 3.129 0.737 3.604 0.828 CGARCH(1,1) 2.258 0.513 2.509 0.572 2.918 0.752 3.347 0.830 APARCH(1,1) 2.245 0.501 2.591 0.568 3.109 0.798 3.583 0.836 GARCHX(1,1,1) 1.843*** 0.437*** 2.137*** 0.514*** 2.485*** 0.661*** 2.788*** 0.770***

Table 7. – This table displays the empirical loss coefficients of various GARCH-type models of the S&P 500 index at multiple horizons under

different variance proxies. For the one-step, five-steps, 21-steps, and 63-steps ahead direct forecasts, we obtained 3,078, 3,074, 3,058, and

3,016 out-of-sample observations, respectively. Note that RMSE coefficients have been scaled up for convenience sake. *** indicates the

preferred forecasting model that yields the lowest loss function coefficient, and thus illustrates the best out-of-sample forecast performance

in this sample.

In contrast to the S&P 500 predictions, the forecast results of HSI, IPC, and KOSPI display a slightly

different picture (see Table B2, Table B3, and Table B4 from the Appendix). To make the results

more concise and accessible for the reader, Table 8 summarises the forecast results of the above-

mentioned tables in which only the outperforming models are demonstrated for each horizon. Purely

observing the loss coefficients of these tables, we conclude that CGARCH(1,1) tends to outperform

GARCHX(1,1,1) in most cases, whilst APARCH(1,1) is superior in only a few cases.


32

These outcomes could possibly be explained by the VIX component in GARCHX, which is closely

linked with the implied volatilities of the S&P 500. This covariate seems to explain the dispersion of

S&P 500 returns more accurately than the asset return volatilities of other equity indices, and there-

fore underperforming models like CGARCH and APARCH in some cases. Table B5 confirms this con-

jecture by showing the adjusted 𝑅2 of the out-of-sample regressions of various volatility proxies of

the indices on the VIX variance proxy, suggesting that the VIX component explains a larger proportion

of the variability in S&P 500 variance proxies than the ones of HSI, IPC, and KOSPI.

Table 8. – Summary of forecast models with the lowest loss coefficients

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) S&P 500 GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX HSI CGARCH GARCHX CGARCH GARCHX CGARCH CGARCH CGARCH GARCHX IPC APARCH CGARCH CGARCH APARCH CGARCH CGARCH CGARCH APARCH KOSPI APARCH CGARCH APARCH CGARCH CGARCH CGARCH CGARCH CGARCH

Panel B: RV (10-min) S&P 500 GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX HSI CGARCH GARCHX CGARCH GARCHX CGARCH CGARCH CGARCH GARCHX IPC APARCH CGARCH CGARCH CGARCH CGARCH CGARCH CGARCH APARCH KOSPI APARCH CGARCH APARCH CGARCH CGARCH CGARCH CGARCH CGARCH

Panel C: RP (Eq. 2) S&P 500 GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX HSI CGARCH GARCHX CGARCH GARCHX CGARCH CGARCH CGARCH GARCHX IPC APARCH APARCH CGARCH APARCH CGARCH CGARCH CGARCH APARCH KOSPI APARCH CGARCH APARCH CGARCH CGARCH CGARCH CGARCH CGARCH

Panel D: RP (Eq. 3) S&P 500 GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX GARCHX HSI CGARCH GARCHX CGARCH GARCHX CGARCH CGARCH CGARCH GARCHX IPC CGARCH CGARCH CGARCH CGARCH CGARCH CGARCH CGARCH APARCH KOSPI APARCH CGARCH APARCH CGARCH CGARCH CGARCH CGARCH CGARCH

Panel E: Statistics CGARCH 5 7 8 6 12 12 12 4 APARCH 7 1 4 2 − − − 4 GARCHX 4 8 4 8 4 4 4 8

Table 8. – This table summarises the forecast results of Table 7, Table B2, Table B3, and Table B4 by showing only the models with the

lowest RMSE and QLIKE losses. Further, Panel E counts the number of times a model is mentioned in the corresponding column. CGARCH,

GARCHX, and APARCH models are mentioned 66, 44, and 18 times in total, respectively. By further decomposing these statistics into four

horizons, we gain additional insights. For one-step ahead, CGARCH, GARCHX, and APARCH are mentioned 12, 12, and 8 times, respectively.

For one-week ahead, CGARCH, GARCHX, and APARCH are mentioned 14, 12, and 6 times, respectively. For one-month ahead, CGARCH and

GARCHX are mentioned 24 and 8 times, respectively. Lastly, for one-quarter ahead, CGARCH, GARCHX, and APARCH are mentioned 16, 12,

and 4 times, respectively.

Besides comparing forecast performances of models, it is also interesting to examine the average

losses across different equity indices at multiple horizons, because it may shed some light on the dis-

similarities of volatility forecasts between developed markets and emerging markets. Hence, Table 9

illustrates the average forecast losses of the four indices for each horizon and variance proxy. We


33

observe that the mean forecast losses of S&P 500 tend to be lower than other indices for all cases

under RMSE, whereas the Hang Seng Index tends to have the highest mean forecast loss compared to

the rest. Nonetheless, the QLIKE-based mean losses are mixed for each horizon. For instance, indices

from emerging markets tend to have the lowest mean forecast losses at longer horizons, but only

occasionally at short horizons. Meanwhile, the S&P 500 has the lowest mean losses only at ℎ = 1 and

ℎ = 5 in Panels A and B under QLIKE. However, the QLIKE-based mean losses of S&P 500 deteriorate

significantly at longer horizons. Ultimately, the results below are rather ambiguous, since it is not

apparent whether volatility forecasts of developed markets indices are likely to have lower losses

compared to emerging markets indices, or vice versa. Therefore, further research may provide clearer

picture on this, but that is beyond the scope of this paper.

Table 9. – Average forecast losses

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) S&P 500 2.785 0.454 3.163 0.536 3.669 0.727 4.098 0.879 HSI 4.664 0.598 4.922 0.623 5.236 0.665 5.615 0.688 IPC 4.649 0.565 4.847 0.602 5.141 0.662 5.361 0.732 KOSPI 3.407 0.565 3.697 0.699 4.261 0.716 4.637 0.686

Panel B: RV (10-min) S&P 500 2.781 0.472 3.143 0.550 3.661 0.742 4.089 0.882 HSI 4.695 0.624 4.962 0.627 5.276 0.696 5.658 0.701 IPC 4.539 0.535 4.768 0.575 5.111 0.643 5.360 0.721 KOSPI 3.473 0.583 3.779 0.682 4.300 0.716 4.648 0.694

Panel C: RP (Eq. 2) S&P 500 2.876 0.629 3.129 0.699 3.585 0.892 4.031 0.995 HSI 5.052 0.771 5.334 0.812 5.677 0.913 6.088 0.880 IPC 4.536 0.571 4.721 0.611 5.066 0.686 5.347 0.765 KOSPI 3.857 0.713 4.202 0.861 4.758 0.859 5.059 0.854

Panel D: RP (Eq. 3) S&P 500 2.906 0.626 3.190 0.682 3.578 0.842 3.958 0.942 HSI 5.097 0.745 5.329 0.782 5.641 0.805 6.001 0.848 IPC 4.470 0.571 4.684 0.606 4.996 0.666 5.210 0.738 KOSPI 3.900 0.682 4.212 0.892 4.778 0.870 5.063 0.835 Table 9. – This table displays the average forecast losses across different equity indices at multiple horizons. The averages are based on the

loss coefficients of GARCH-type models in Table 7, Table B2, Table B3, and Table B4.

Note that the reported forecast losses vary moderately under the 5-minute and 10-minute subsam-

pled realised volatility proxies and the two range-based variance proxies. It is important to use sev-

eral unbiased variance estimators when evaluating the validity and reliability of out-of-sample fore-

casts, because consistency between variance estimators leads to better understanding of a model’s

predictive ability. Therefore, excluding a variance proxy because of unfavourable forecast losses may

potentially lead to unrepresentative conclusions.


34

Figure 7. – One-step ahead volatility forecasts of S&P 500

Figure 7. – This figure illustrates the rolling one-step ahead volatility forecasts of ARCH(1), GARCH(1,1), and GJR-GARCH(1,1) models. Each

panel illustrates the in- and out-of-sample conditional variances, which is separated by the grey dashed line. The in-sample period starts from

January 5, 2000, to August 6, 2004, resulting in 1,000 observations, whereas the forecast period starts from August 9, 2004, to March 27, 2018,

resulting in 3,078 observations.


35

Figure 8. – One-step ahead volatility forecasts of S&P 500

Figure 8. – This figure illustrates the rolling one-step ahead volatility forecasts of CGARCH(1), APARCH(1,1), and GARCHX(1,1,1) models. Each

panel illustrates the in- and out-of-sample conditional variances, which is separated by the grey dashed line. The in-sample period starts from

January 5, 2000, to August 6, 2004, resulting in 1,000 observations, whereas the forecast period starts from August 9, 2004, to March 27, 2018,

resulting in 3,078 observations.


36

• 5.2.2. Results of Diebold-Mariano tests

To check whether the forecast results under the RMSE and QLIKE loss functions are statistically

significant, we conduct Diebold-Mariano tests for only the S&P 500 in order to perform pairwise com-

parisons of GARCH-type models. The DM test statistics for each horizon, which are obtained by re-

gression of 𝑑𝑖𝑗,𝑡 on a constant with HAC standard errors, are reported in the Appendix section; see

Table B6, Table B7, Table B8, and Table B9. From these tables, we observe positive and negative

DM test statistics. A positive DM test statistic means that the column model tends to be less accurate

compared to the corresponding row model, whereas a negative DM test statistic implies that the col-

umn model tends to predict more accurately than the corresponding row model.

According to these four tables, ARCH(1) seems to be the least desired model for volatility forecast-

ing, as other GARCH models tend to generate more accurate forecasts. Further, the test statistics of

medium-term horizons, ℎ > 5, under QLIKE tend to be less significant than the ones of short-term

horizons, suggesting that medium-term forecast outperformance of a pair of models happens mostly

by chance. In other words, the two competing models are inclined to perform equally well, or bad,

under QLIKE when it comes to one-month or one-quarter ahead forecasts of conditional variances of

the S&P 500. This is obviously not remarkable due to the uncertain nature of long-term forecasts.

What is striking is that for shorter horizons the combinations of APARCH(1,1) and GJR-GARCH(1,1),

and CGARCH(1,1) and GJR-GARCH(1,1) models tend to be statistically insignificant under both RMSE

and QLIKE. Note that the GJR-GARCH model can be derived from APARCH, since it is a special case of

the latter. This fact may explain to some extent the insignificant DM test statistics of the APARCH/GJR-

GARCH combination.

Nonetheless, by placing each forecast model in juxtaposition as a pair, we cannot effortlessly point

out which forecast models are superior to other forecast models for all horizons. In fact, the Diebold-

Mariano test does not provide clear answers to the overall performance of models for a specific hori-

zon. Therefore, the use of Model Confidence Sets is more convenient in this case.

• 5.2.3. Results of MCS

This paper uses the “MCS” package in R to determine the MCS 𝑝-values.13 We use the block boot-

strap procedure and set the parameters to 𝐵 = 1,000 resamples, block length ℒ = ℎ, and significance

level 𝛼 = 5%. The block length of this bootstrap procedure not only depends on the horizon of our

forecast losses, but also on the potential serial correlation of these losses. Thus, the larger the forecast

horizon, the larger the block length of the bootstrap. Note that we set block length ℒ ≠ ℎ only for the

13 See Catania and Bernardi (2017) for documentation of this R package.


37

one-step ahead forecasts, in order to take into account any possible serial correlated forecast errors.

Hence, for ℎ = 1, we set ℒ = 2. The MCS results of the S&P 500 are reported in Table 10, whereas the

results of the remaining equity indices are illustrated in the Appendix, as usual. Forecast models that

are included in ℳ̂0.95∗ , or 95% Model Confidence Set, are identified by asterisks.

Table 10. – Results of MCS for S&P 500

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0.488*** 0 0.679*** GJR-GARCH(1,1) 0 0 0 0 0 0.169*** 0 0.478*** CGARCH(1,1) 0 0 0 0 0 0.084*** 0 0.350*** APARCH(1,1) 0 0 0 0 0 0.185*** 0 0.548*** GARCHX(1,1,1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000***

Panel B: RV (10-min) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0.431*** 0 0.595*** GJR-GARCH(1,1) 0 0 0 0 0 0.155*** 0 0.483*** CGARCH(1,1) 0 0 0 0 0 0.091*** 0 0.284*** APARCH(1,1) 0 0 0 0 0 0.149*** 0 0.521*** GARCHX(1,1,1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000***

Panel C: RP (Eq. 2) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0.206*** 0 0.394*** GJR-GARCH(1,1) 0 0 0 0 0 0.087*** 0 0.359*** CGARCH(1,1) 0 0 0 0 0 0.107*** 0 0.271*** APARCH(1,1) 0 0 0 0 0 0.084*** 0 0.296*** GARCHX(1,1,1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000***

Panel D: RP (Eq. 3) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0.052*** 0 0.105*** GJR-GARCH(1,1) 0 0 0 0 0 0.057*** 0 0.108*** CGARCH(1,1) 0 0 0 0 0 0.115*** 0 0.077*** APARCH(1,1) 0 0 0 0 0 0.050*** 0 0.079*** GARCHX(1,1,1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000***

Table 10. – This table provides the 𝑝-values of Model Confidence Sets for the S&P 500 forecast losses at multiple horizons. A block bootstrap

approach of 1,000 resamples, block length equal to forecast horizon, and 5% significance level are used to determine these coefficients. Note

that we set ℒ = 2 for the one-step ahead forecasts. *** denotes the forecast model is part of ℳ̂0.95∗ .

From the table above, GARCHX seems to be the only model included in ℳ̂0.95∗ for all horizons under

RMSE, whereas ARCH is the only model excluded from ℳ̂0.95∗ for ℎ = 21 and ℎ = 63 under QLIKE. In

contrast to Quaedvlieg’s (2018) results, the QLIKE-based MCS for the S&P 500 expands for ℎ > 5 in

this paper. At first, the results of this study seem conflicting, but there may be a logical explanation

for these findings. It could be that due to the likelihood of increased forecast uncertainty at longer

horizons, various GARCH models are considered to be superior in this case. In other words, they tend

to perform equally well for longer horizons. Altogether, the results of Table 10 seem to suggest that


38

the customised GARCHX model is a superior model for both short- and medium-term forecasting of

conditional variances of the S&P 500. Again, it seems that adding an accurate variance proxy to the

standard GARCH model improves the predictive ability in this case.

Furthermore, Table B10 reports the MCS results related to the Hang Seng Index. Here we observe

a slightly different picture in which CGARCH tends to be the sole superior model for all horizons under

RMSE. Although the component GARCH is included in ℳ̂0.95∗ for all horizons under QLIKE, the custom-

ised GARCHX model is ranked higher in terms of MCS 𝑝-value in most cases. It is remarkable that

ARCH is part of the 95% MCS in Panels B and C for ℎ = 21 under QLIKE, since the forecast errors of

ARCH tend to be relatively large compared to other models, see previous subsubsections. Despite this,

the CGARCH and GARCHX models are mostly preferred due to the ranking of MCS 𝑝-values.

Results related to the Mexican IPC index are reported in Table B11, where only the component

GARCH model is part of ℳ̂0.95∗ for almost all horizons under RMSE. However, when using either the

10-minute subsampled realised variance proxy or the range proxy based on Eq. (2), the APARCH

model is preferred for one-step ahead forecasts under RMSE. It seems that ℳ̂0.95∗ tends to expand over

the horizons under the QLIKE loss function, including forecast models such as CGARCH, APARCH, GJR-

GARCH, GARCHX, and even GARCH in ℳ̂0.95∗ . Altogether, depending on the loss functions, the results

suggest that CGARCH and APARCH, and sometimes GJR-GARCH and GARCHX as well, are the most

preferred models for volatility forecasting, since they are ranked highest in terms of MCS 𝑝-value.

Lastly, Table B12 illustrates the results of MCS for the Korea Composite Stock Price Index. In this

case, the APARCH model tends to be the sole superior model for short horizons under RMSE, while

the component GARCH model seems to be the only model in ℳ̂0.95∗ for longer horizons under the same

loss function. Based on the QLIKE loss function, almost all models are included in ℳ̂0.95∗ across the

horizons. Models that are frequently included in the 95% MCS are CGARCH, APARCH, GJR-GARCH,

and GARCHX. For some horizons under QLIKE, the simple GARCH(1,1) model or even ARCH(1) are

included as well. As a whole, both APARCH and CGARCH are preferred for shorter horizons, while

CGARCH and GARCHX tend to be better models for long-term forecasting.


39

§6. Research limitations

Like all other research studies, this paper has its limitations in terms of empirical research as well.

Therefore, it is important to address several of those limitations here. Firstly, the split of sample data

into an in-sample and out-of-sample period is done arbitrarily in this study, i.e. without additional

robustness tests. The current approach in this paper might affect the out-of-sample forecast results

to a certain extent. Whether this has large implications for the validity of this empirical study is not

entirely clear yet. On the other hand, finance literature has hitherto not synthesised a proper meth-

odological guidance on the issue of selecting the right split point that separates the sample data into

two parts for forecasting purposes (Hansen & Timmermann, 2012).

Secondly, this paper did not use the iterated approach next to the direct method for out-of-sample

forecasts, since the former approach requires advanced programming skills. Using both methods

would make a comparison between competing forecast models complete, in the sense that one should

not only focus on the prediction models, but also on the forecast procedure itself. Unfortunately, the

“rugarch” package in R does not have a built-in function yet for computing iterated forecasts, hence

leaving it as a cumbersome programming exercise to the researcher.

Moreover, the model parameters were estimated under the classic Gaussian distribution, which is

often criticised in finance literature. Mandelbrot (1963), Fama (1965), and Poon and Granger (2005)

have shown evidence that the returns of financial assets tend to follow a non-normal distribution.

Despite its convenience, the Gaussian distribution does not appropriately capture the heavy tails of

asset return distributions (Christoffersen, 2012). As a result, it understates the probability of extreme

changes in asset returns (Hull, 2015), which eventually affects the forecast accuracy of models.

Finally, this paper evaluated the direct forecast results of GARCH-type models individually at each

horizon. It does not use specific evaluation tests for joint multistep testing of out-of-sample forecasts,

due to lack of popular joint multi-horizon forecast evaluation tests in finance literature. Despite this,

Quaedvlieg (2018) proposed an extension to the MCS approach of Hansen et al. (2011), which allows

for joint multi-horizon testing of forecast models at once. A potential drawback of evaluating multi-

horizon forecast performances of various models individually is that it may potentially lead to spuri-

ous outcomes, and thus resulting in faulty conclusions about the predictive ability of forecast models.

Accordingly, caution should be exercised when examining the empirical results of volatility fore-

casts in this research paper due to its limitations.


40

§7. Suggestions for further research

Throughout the years, several empirical studies have conducted research on volatility-forecasting

models and their performances. Many have tried to come up with novel mathematical models and

better proxy measures of market volatility to improve the predictive power of models so that more

accurate trajectory of future dispersions of asset returns can be forecasted. Regardless of all the em-

pirical studies that have been done in the past, research in volatility forecasting is still far from com-

plete due to the complex nature of this topic. In addition, advances in the field of computer science

may potentially lead to new applications and techniques for volatility-forecasting research, which

might lead to new insights. Hence, further research in this field is still relevant.

For instance, one could use more complex univariate GARCH-type models, such as the Fractionally

Integrated GARCH (FIGARCH) or Fractionally Integrated APARCH (FIAPARCH) models. Additionally,

one could extend the GARCHX model by adding additional external regressors, such as a covariate

that captures volatility-spillover effects from other financial markets (Diebold & Yilmaz, 2009), or

volatility of volatility measure of an underlying equity index, e.g. Cboe VVIX Index, to pick up early

signals of market volatility.14 Furthermore, one could also evaluate forecasting performances of mul-

tivariate GARCH-type models at multiple horizons. This is interesting for investment managers who

manage large multi-asset funds since they invest across a number of different asset classes, such as

cash equities and FICC-related instruments. When it comes to both active and passive management

of portfolios, multivariate forecast models could help asset managers with their decisions on the

funds’ strategic and tactical asset allocation.

From a theoretical point of view, researchers could focus on novel ways to improve the computa-

tional efficiency of estimations of nonlinear model parameters. Any improvement in computational

speed can potentially lead to a short-term competitive edge for high-frequency traders who are spe-

cialised in volatility-linked exchange traded products. Moreover, to make well-informed decisions

based on multi-horizon forecasts, it is necessary to have common measures of predictive ability to

evaluate direct or iterated forecasts at multiple horizons jointly rather than considering them indi-

vidually. Because evaluating multi-horizon forecasting performances of various models individually

may potentially lead to spurious results, and thus premature conclusions (Quaedvlieg, 2018).

14 According to Cboe’s website, the VVIX is a volatility of volatility measure in that it represents the expected volatility of the 30-day forward price of the VIX.


41

§8. Summary and conclusion

This paper evaluates the multi-horizon forecast performances of six GARCH-type models, namely

the ARCH, GARCH, GJR-GARCH, CGARCH, APARCH, and GARCHX models, using four equity indices of

which two developed markets indices (S&P 500 and HSI) and two emerging markets indices (IPC and

KOSPI). Additionally, we use several variance proxies, namely realised volatilities and range-based

volatilities, to assess the forecast performances of these GARCH-type models. We evaluate the fore-

cast performances with MSE and QLIKE loss functions. Further, we introduce the Diebold-Mariano

test statistic (Diebold & Mariano, 1995) and Model Confidence Set (Hansen et al., 2011) to test the

significance of forecast losses at multiple horizons. The aim of this paper is to find out which forecast

models generate the most accurate out-of-sample direct forecasts of conditional variances in the

short run and in the medium term. In addition, this paper studies the underlying features that drive

the performance of GARCH models at multiple horizons. Thus, the research questions are (1) “Which

of the following GARCH-type models generate the best volatility forecasts in the short term: ARCH,

GARCH, GJR-GARCH, CGARCH, APARCH, or GARCHX?”, (2) “Which of the following GARCH-type models

generate the best volatility forecasts in the medium term: ARCH, GARCH, GJR-GARCH, CGARCH, APARCH,

or GARCHX?”, and (3) “What key features drive the forecast performance of these GARCH-type models

at multiple horizons?”

To answer the research questions properly, we cannot draw general conclusions for all equity indi-

ces based on the forecast results due to the variations in variance proxies, forecast horizons, loss

functions, and significance tests. Therefore, we need to answer these questions independently for

each equity index. For the S&P 500, the undisputed forecast model for both short and longer horizons

is the customised GARCHX(1,1,1) model, which incorporates an external regressor based on the VIX

Index, irrespective of the loss function. Although the 95% MCS of the S&P 500 includes other forecast

models as well for longer horizons under QLIKE, GARCHX is still ranked the highest in terms of MCS

𝑝-value. For the Hang Seng Index, depending on the loss function, CGARCH and GARCHX perform well

when it comes to short- and medium-term volatility forecasts. Both models are frequently included

in ℳ̂0.95∗ under QLIKE. Meanwhile, RMSE-based ℳ̂0.95

∗ considers CGARCH as the sole superior model.

In addition, for IPC, we observe mixed results based on the Model Confidence Sets. Under RMSE, the

CGARCH is considered as the sole superior model for all horizons in almost all cases, except for the

10-minute subsampled realised proxy and the range proxy based on Eq. (2) at ℎ = 1, where APARCH

is part of ℳ̂0.95∗ . However, under QLIKE loss function, a variety of models are selected for the 95%


42

MCS of IPC at multiple horizons, including CGARCH, APARCH, GJR-GARCH, and GARCHX. Still, the 𝑝-

value ranks the first two models as highest in almost all cases. Based on the results of forecast losses

and depending on the loss function, the component GARCH model tend to generate the lowest forecast

errors in most cases. Although in some cases, the APARCH model performs better. For KOSPI, the

results of forecast losses favour CGARCH for all horizons under QLIKE, while under RMSE APARCH

tends to be the better model for short-term forecasts but is surpassed by CGARCH at longer horizons.

Based on the MCS results, APARCH tends to be the sole superior model for short horizons under

RMSE, whereas CGARCH seems to be the only model in ℳ̂0.95∗ for horizons larger than a week. Under

QLIKE however, many models are included in ℳ̂0.95∗ for all horizons.

Ultimately, we can conclude that the empirical results of the conditional variance forecasts of these

equity indices are rather mixed, since there is no generic forecast model that shows superior perfor-

mance in every case for all financial assets. Thus, the accuracy of volatility forecasts are mainly de-

pendent upon factors such as forecast procedure, type of variance estimators, loss functions, signifi-

cance tests, and even underlying data. Yet, from the results of forecast losses, we can conclude that

one model consistently underperforms other GARCH models for all equity indices, namely ARCH(1).

This can be explained by the fact that the conditional variance equation of ARCH is simply a linear

function of past errors that does not incorporate past conditional variances (Engle, 1982). However,

in some cases, the uncomplicated ARCH model is part of ℳ̂0.95∗ , but it does not necessarily mean that

it is preferred over other GARCH-type models when it comes to volatility forecasting.

Altogether, we can conclude that several factors affect the forecast performance of GARCH models.

For instance, adding an external regressor to the GARCH(1,1) model significantly improves the con-

ditional variance forecasts for the S&P 500, but not necessarily for IPC and KOSPI. Admittedly, we

extended the standard GARCH model only with the VIX variance proxy and not with similar volatility

indices for the other equity indices, resulting in GARCHX being surpassed by APARCH and CGARCH in

terms of forecast performance. In addition, the structure of a conditional variance equation affect the

out-of-sample volatility forecasts to some extent. For example, the flexible conditional variance equa-

tion of APARCH provides relatively accurate forecasts compared to GARCH(1,1), since APARCH is able

to capture empirical regularities related to asset return volatility and nests multiple GARCH-type

models (Laurent, 2004). Moreover, the CGARCH decomposes volatility into a long-run component and

a short-run component (McMillan & Speight, 2004), allowing the conditional variance to mean revert

to a time-varying long-run variance rate (Engle & Lee, 1999). Hence, this could possibly explain the

forecast results of APARCH and CGARCH for some emerging markets indices in this paper.


43

This study adds to existing literature by comparing multi-horizon forecast performances of GARCH-

type models based on equity indices from developed markets and emerging markets. Additionally,

this paper uses several variance proxies for evaluation of forecast results, such as the 5-minute and

10-minute subsampled realised proxies and two range-based proxies. Furthermore, it examines the

underlying features of GARCH models that drive the forecast performance at different horizons.

More importantly, what are the implications of this study? First, finance practitioners, such as quant

traders, investment managers, and risk managers, should work with a framework where one con-

ducts volatility forecasts and assesses forecast performances in a holistic manner by using multiple

GARCH models, forecast approaches, variance proxies, loss functions, and significance tests. This is

necessary due to the variations in forecast performances and likelihood of conflicting results. Thus,

this framework can provide a comprehensive overview of forecast results to finance practitioners so

that they can make deliberate decisions amidst severe market turmoil. Second, to make the most of

mixed forecast results, one could use forecast combinations next to individual forecasts, since com-

bining predictions leads to possible diversification benefits. Compared to individual forecasts, fore-

cast combinations are more robust against misspecification biases and measurement errors in the

datasets underlying the individual forecasts (Timmermann, 2006). Third, adding the VIX to a forecast

model does not necessarily improve the forecast performance for other equity indices. Therefore, it

is essential to include an index-specific volatility measure to the model.

Despite the comprehensive study on GARCH-type models, this paper, unfortunately, has its research

limitations as well. For instance, the data sample split is done randomly without any further robust-

ness tests. We also did not use the iterated approach for estimations of conditional variances. Further,

the model parameters are all estimated under the normal distribution. Additionally, we evaluated the

direct forecasts of GARCH-type models individually at each horizon. This paper does not use specific

tests in order to test multistep out-of-sample forecasts jointly at once. Hence, these limitations may

have resulted in a one-sided view on the evaluation of forecast performances.

Finally, further research is still relevant in this particular field due to the complexity of this topic.

For instance, one could use convoluted nonlinear GARCH models and even extend them with covari-

ates that may improve the models’ predictive ability. Additionally, evaluating multivariate GARCH

models may provide unique solutions to issues related to multi-asset portfolio management, which is

beneficial to institutional investors. Besides, one could focus on novel estimation methods and tech-

niques to improve the computational efficiency of model parameter estimates, or on evaluation tests

that allows for joint multi-horizon testing of forecast models at once.


44

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49

Appendix

Section A – Significance tests

• A.1. Jarque-Bera normality test

To find out whether a statistical dataset is normally distributed, one can conduct tests for normality

of observations such as the Jarque-Bera test (Jarque & Bera, 1987). The null hypothesis, 𝐻0, of this

test states that the sample data is normally distributed, i.e. a joint hypothesis in which skewness and

kurtosis coefficients are equal to zero and three, respectively. Meanwhile, the alternative hypothesis,

𝐻𝑎, states that the sample data is not normally distributed. As described in Jarque and Bera (1987),

the Lagrange multiplier (LM) test statistic, or Jarque-Bera test statistic, is defined by

𝐿𝑀 = 𝑁 ((√𝑏1)

2

6+

(𝑏2 − 3)2

24), (A.1)

where 𝑁 is the number of sample observations, and √𝑏1 and 𝑏2 are the sample skewness and kurtosis

coefficients, respectively. It turns out that this test statistic can be compared asymptotically with a

chi-squared, 𝜒2, distribution with two degrees of freedom. For large samples, 𝐻0 is rejected if the LM

test statistic exceeds a critical value from the 𝜒(2)2 distribution (Jarque & Bera, 1987). These critical

values can be easily obtained from online sources.15 With two degrees of freedom, the 1% significance

level results in a critical value of 9.21.

• A.2. Augmented Dickey-Fuller test

An issue that frequently occurs in time series analysis is the occurrence of nonstationary processes

in economic time series. To examine whether unit roots are present in economic time series, the aug-

mented Dickey-Fuller (ADF) test can be conducted (Dickey & Fuller, 1979). The ADF test can be de-

scribed in two steps. First, consider a simple first order autoregressive (AR) model

𝑥𝑡 = 𝜌𝑥𝑡−1 + 휀𝑡 , (A.2)

where 𝑥𝑡 stands for value of series in period 𝑡, while 𝜌 is a coefficient. A unit root is present in a time

series when 𝜌 = 1, which implies that the series follows a nonstationary process (Harris, 1992). Sta-

tionary time series are usually obtained by simply taking the first order, or even second order, differ-

ence of the variable of interest. Hence, the next step of the ADF test is

Δ𝑥𝑡 = 𝜙𝑥𝑡−1 + 휀𝑡 , (A.3)

15 𝜒2 distribution table source: “http://www.socr.ucla.edu/Applets.dir/ChiSquareTable.html”.


50

which is the first order difference. In this case, the 𝐻0 of the ADF test states that a unit root is present

if 𝜙 = 0, which is equivalent to testing 𝜌 = 1. The alternative hypothesis is 𝜙 < 0, which states that a

unit root is not present, i.e. the variable of interest follows a stationary process.

The results of the unit root tests are illustrated in Table A1, in which sample data of equity indices

are tested on both index price level and return level. The ADF test coefficients from Panel A show

insignificant results for the two Asian equity indices, whereas the two indices from the Americas are

statistically significant at the 10% level. Hence, we fail to reject the null hypothesis for the Asian indi-

ces, suggesting that these equity indices tend to follow a unit root process on an index price level.

Furthermore, Panel B displays the ADF test coefficients of logarithmic returns. All of them are statis-

tically significant at the 1% level, which implies that the logarithmic returns of these stock indices

tend to follow a stationary process.

Table A1. – Unit root tests for price levels and log returns

Indices Coefficient Std. error 𝒕-stat 𝒑-value

Panel A: Price level S&P 500 -0.002* 0.001 -1.853 0.064 HSI -0.001 0.002 -0.463 0.643 IPC -0.003* 0.002 -1.680 0.093 KOSPI -0.002 0.002 -1.451 0.147

Panel B: Log returns S&P 500 -1.092*** 0.017 -66.133 0 HSI -1.059*** 0.016 -68.238 0 IPC -0.929*** 0.016 -57.487 0 KOSPI -1.018*** 0.016 -62.674 0

Table A1. – This table illustrates the ADF test coefficients of the stock indices based on price levels

and log returns, which are represented in Panels A and B, respectively. In total, 2,948 observations

were used in this test. *** and * denote significance at 1% and 10% level, respectively.

In addition, we perform the augmented Dickey-Fuller test on the volatility proxies of these stock

indices as well, which are reported in Table A2. From this table, we observe statistically significant

ADF coefficients at the 1% significance level in each panel. These results suggest that there is suffi-

cient statistical evidence to reject the null hypothesis for all variance proxies in every case. Hence, the

variance proxies tend to follow a stationary process because we do not find statistical evidence of the

presence of unit roots in these series.


51

Table A2. – Unit root tests for volatility proxies

Indices Coefficient Std. error 𝒕-stat 𝒑-value

Panel A: RV (5-min) S&P 500 -0.154*** 0.014 -10.799 0 HSI -0.380*** 0.011 -33.297 0 IPC -0.586*** 0.017 -35.311 0 KOSPI -0.262*** 0.013 -19.919 0

Panel B: RV (10-min) S&P 500 -0.173*** 0.015 -11.715 0 HSI -0.384*** 0.011 -33.562 0 IPC -0.476*** 0.018 -27.175 0 KOSPI -0.303*** 0.014 -21.266 0

Panel C: RP (Eq. 2) S&P 500 -0.411*** 0.014 -29.993 0 HSI -0.551*** 0.010 -55.733 0 IPC -0.532*** 0.014 -37.981 0 KOSPI -0.332*** 0.016 -21.021 0

Panel D: RP (Eq. 3) S&P 500 -0.350*** 0.016 -22.084 0 HSI -0.715*** 0.012 -60.927 0 IPC -0.492*** 0.014 -35.290 0 KOSPI -0.374*** 0.014 -26.338 0

Table A2. – This table illustrates the ADF test coefficients of the 5 and 10-minutes realised vari-

ances and range-based variances of the four equity indices. In total, 2,948 observations were used

in this test. *** denotes significance at 1% level.

• A.3. Ljung-Box portmanteau test

Qualitative tools such as the sample autocorrelation and partial autocorrelation functions can easily

assess the presence of serial correlation at individual lags. On the other hand, quantitative tools like

the Ljung-Box portmanteau test, or Q-test, can be used for testing serial correlation jointly at multiple

lags, i.e. it tests the overall randomness of series based on a number of lags instead of testing random-

ness at each distinct lag (Ljung & Box, 1978). The null of Ljung-Box Q-test states that the first 𝓀 serial

correlations of a series are jointly equal to zero, which means that the series is independently distrib-

uted. In other words, 𝐻0: 𝜌1 = 𝜌2 = ⋯ = 𝜌𝓀 = 0. Meanwhile, the alternative hypothesis states that

the series exhibits autocorrelation. The Q-test statistic is formally defined as

�̃�(𝓀) = 𝑛(𝑛 + 2) ∑�̂�𝑚

𝑛 − 𝑚

𝓀

𝑚=1

, (A.4)

where 𝑛 is the sample size, �̂�𝑚 is the sample autocorrelation at lag 𝑚, and 𝓀 is the number of lags

being tested. When the sample size is sufficiently large, �̃� could be distributed approximately as 𝜒(𝓀)2

with 𝓀 degrees of freedom (Box & Pierce, 1970). In this case, the null is rejected with significance

level 𝛼 if the Ljung-Box �̃� > 𝜒(𝓀),1−𝛼2 , in which 𝜒(𝓀),1−𝛼

2 is the 1 − 𝛼 quantile of the 𝜒(𝓀)2 distribution.


52

In Table A3, we observe the Ljung-Box Q-test statistics of the variance estimators. We can conclude

that there is sufficient statistical evidence to reject the null hypothesis for all cases, which indicates

that these volatility proxy series tend to be serially correlated. In accordance with the autocorrelation

and partial autocorrelation plots from Section 4.1, the Ljung-Box Q-test confirms the occurrence of

serial correlation in these series.

Table A3. – Ljung-Box test statistics

Lag = 1

Lag = 20

Lag = 40

Indices �̃�(𝓴 = 𝟏) 𝒑-value �̃�(𝓴 = 𝟐𝟎) 𝒑-value �̃�(𝓴 = 𝟒𝟎) 𝒑-value

Panel A: RV (5-min) S&P 500 1,041.86*** 0 9,641.91*** 0 14,777.99*** 0 HSI 1,116.47*** 0 8,443.70*** 0 12,823.92*** 0 IPC 278.00*** 0 2,858.52*** 0 4,011.57*** 0 KOSPI 1,357.33*** 0 11,293.16*** 0 15,761.83*** 0

Panel B: RV (10-min) S&P 500 1,022.73*** 0 9,464.34*** 0 14,578.30*** 0 HSI 1,049.04*** 0 7,580.12*** 0 11,435.60*** 0 IPC 348.04*** 0 3,906.69*** 0 5,604.53*** 0 KOSPI 1,198.31*** 0 10,030.04*** 0 14,046.23*** 0

Panel C: RP (Eq. 2) S&P 500 880.63*** 0 8,271.80*** 0 12,941.21*** 0 HSI 629.84*** 0 3,221.93*** 0 4,749.52*** 0 IPC 537.13*** 0 4,941.63*** 0 7,449.07*** 0 KOSPI 1,012.11*** 0 5,457.85*** 0 7,370.89*** 0

Panel D: RP (Eq. 3) S&P 500 752.20*** 0 6,659.52*** 0 10,328.59*** 0 HSI 261.47*** 0 2,473.92*** 0 3,828.09*** 0 IPC 586.12*** 0 4,806.33*** 0 7,145.75*** 0 KOSPI 1,156.94*** 0 5,260.44*** 0 6,980.69*** 0 Table A3. – This table illustrates the results of Ljung-Box Q-test statistics of the volatility proxies. We use three different

lag periods in this test, namely one lag, 20 lags, and 40 lags. The critical values corresponding to one, 20, and 40 degrees of

freedom at 1% significance level from a 𝜒(𝓀)2 distribution are 6.635, 37.566, and 63.691, respectively. Furthermore, *** de-

notes significance at 1% level.

• A.4. Engle’s ARCH test

A time series revealing serial correlation in the disturbances, also known as conditional heteroske-

dasticity, is said to have ARCH effects. This phenomenon causes uncorrelated time series to be serially

dependent because of a dynamic conditional variance process. To gauge the significance of ARCH ef-

fects in economic time series, we can use a Lagrange multiplier test called the ARCH test (Engle, 1982).

The null propounds that the squared errors, 𝜉𝑡2, are not serially dependent, i.e. 𝐻0: 𝛼0 = 𝛼1 = ⋯ =

𝛼𝑛 = 0. In contrast, the alternative hypothesis of this test simply states that 𝜉𝑡2 are serially correlated

(Tsay, 2005), given by the following regression model 𝐻𝑎: 𝜉𝑡2 = 𝛼0 + ∑ 𝛼𝑖

𝑘𝑖=1 𝜉𝑡−𝑖

2 + 𝑢𝑡. Engle’s ARCH

test statistic is the common 𝐹-statistic for the regression on squared errors. Under the null, the 𝐹-


53

statistic follows a chi-squared distribution with 𝑚 degrees of freedom (Tsay, 2005). A substantially

large critical value suggest rejection of the null in favour of the alternative hypothesis.

As described in Engle (1982), to obtain the ARCH test statistics, we perform the following steps:

Step I. Regress the log return series on a constant using ordinary least squares (OLS).

Step II. Predict the residuals of the logarithmic return series.

Step III. Regress the predicted residuals on a constant using OLS.

Step IV. Perform the ARCH LM test.

In Table A4, we find the test statistics of Engle’s ARCH LM tests. We use three different lag periods to

check for distant ARCH effects in our time series data. The results below suggest that there is sufficient

statistical evidence to reject the null for some cases. Note that the ARCH test statistics for IPC and

KOSPI indices are statistically insignificant only for the first lag, whereas these ARCH test statistics

tend to be significant after five lag periods. Still, these results suggest that the squared residuals of

these equity indices tend to be serially dependent, which implies that forecast models that allow for

conditional heteroskedasticity could potentially generate more accurate volatility estimates.

Table A4. – ARCH LM test statistics

Lag = 1 Lag = 5 Lag = 10

Indices Test statistic 𝒑-value Test statistic 𝒑-value Test statistic 𝒑-value

S&P 500 8.305*** 0.004 40.076*** 0 46.021*** 0 HSI 4.126*** 0.042 19.303*** 0.017 31.568*** 0.001 IPC 1.072 0.301 38.834*** 0 41.608*** 0 KOSPI 1.959 0.162 13.289*** 0.021 16.485* 0.087 Table A4. – This table illustrates the in-sample results of the ARCH LM test. In this test, we use three different lag periods,

namely one, five, and ten lags. The critical values for one, five, and ten degrees of freedom at 1% significance level from a

𝜒(𝓀)2 distribution are 6.635, 15.086, and 23.209, respectively. Furthermore, *** and * denote the significance at 1% and

10% levels, respectively.


54

Section B – Supplementary tables

Table B1. – In-sample estimates of model parameters

Forecast model 𝝎𝟎 𝜶𝒋 𝝋𝒊 𝝀𝒋 𝒙 𝒚 𝜹 𝝊𝒌

Panel A: S&P 500

ARCH(1) 0

(0)

[0.998]

0.932***

(0.025)

[0]

GARCH(1,1) 0

(0)

[0.931]

0.069

(0.115)

[0.548]

0.928***

(0.112)

[0]

GJR-GARCH(1,1) 0

(0)

[0.950]

0.012

(0.120)

[0.923]

0.938***

(0.174)

[0]

0.096

(0.118)

[0.418]

CGARCH(1,1) 0

(0)

[0.728]

0

(0.146)

[1.000]

0.406

(0.820)

[0.621]

0.996***

(0.009)

[0]

0.067

(0.055)

[0.228]

APARCH(1,1) 0

(0)

[0.271]

0.046***

(0.001)

[0]

0.950***

(0.005)

[0]

1.000***

(0)

[0]

1.129***

(0.198)

[0]

GARCHX(1,1,1) 0

(0)

[1.000]

0.015

(0.020)

[0.464]

0.393

(0.373)

[0.292]

0.458

(0.299)

[0.126]

Panel B: IPC

ARCH(1) 0***

(0)

[0]

0.171*

(0.091)

[0.059]

GARCH(1,1) 0

(0)

[0.980]

0.048

(0.538)

[0.930]

0.945

(0.582)

[0.104]

GJR-GARCH(1,1) 0***

(0)

[0]

0.019

(0.014)

[0.170]

0.857***

(0.021)

[0]

0.181***

(0.062)

[0.004]

CGARCH(1,1) 0

(0)

[0.993]

0.150

(0.339)

[0.658]

0.668***

(0.084)

[0]

1.000***

(0.004)

[0]

0.020

(0.042)

[0.633]

APARCH(1,1) 0.001

(0.002)

[0.581]

0.102***

(0.029)

[0]

0.881***

(0.036)

[0]

0.687***

(0.225)

[0.002]

0.908**

(0.394)

[0.021]

GARCHX(1,1,1) 0

(0)

[0.995]

0.048

(2.085)

[0.982]

0.945

(2.446)

[0.699]

0

(0.198)

[1.000]

Continued on next page


55

Forecast model 𝝎𝟎 𝜶𝒋 𝝋𝒊 𝝀𝒋 𝒙 𝒚 𝜹 𝝊𝒌

Panel C: HSI

ARCH(1) 0***

(0)

[0]

0.144

(0.117)

[0.215]

GARCH(1,1) 0

(0)

[0.702]

0.056

(0.041)

[0.170]

0.924***

(0.022)

[0]

GJR-GARCH(1,1) 0***

(0)

[0]

0

(0.005)

[1.000]

0.919***

(0.009)

[0]

0.105***

(0.030)

[0]

CGARCH(1,1) 0

(0)

[0.994]

0.044

(0.180)

[0.808]

0.903***

[0.082]

[0]

1.000***

(0.007)

[0]

0.016

(0.030)

[0.606]

APARCH(1,1) 0***

(0)

[0]

0.032***

(0.007)

[0]

0.921***

(0.024)

[0]

1.000***

(0.001)

[0]

1.785***

(0.097)

[0]

GARCHX(1,1,1) 0

(0)

[0.845]

0.059

(0.083)

[0.477]

0.912***

(0.162)

[0]

0.014

(0.046)

[0.765]

Panel D: KOSPI

ARCH(1) 0***

(0)

[0]

0.039

(0.047)

[0.407]

GARCH(1,1) 0

(0)

[0.854]

0.011***

(0.002)

[0]

0.986***

(0.003)

[0]

GJR-GARCH(1,1) 0***

(0)

[0]

0

(0.010)

[1.000]

0.945***

(0.011)

[0]

0.075***

(0.026)

[0.003]

CGARCH(1,1) 0

(0)

[0.986]

0.075***

(0.121)

[0.537]

0.822***

(0.031)

[0]

1.000***

(0.001)

[0]

0.011

(0.040)

[0.787]

APARCH(1,1) 0.001

(0.002)

[0.482]

0.074***

(0.025)

[0.003]

0.893***

(0.034)

[0]

0.880**

(0.349)

[0.012]

0.975***

(0.308)

[0.002]

GARCHX(1,1,1) 0

(0)

[0.798]

0.097***

(0.024)

[0]

0.770***

(0.124)

[0]

0.290

(0.255)

[0.256]

Table B1. – This table illustrates the in-sample parameter estimates of the forecast models. Values in parentheses are

robust standard errors while values in brackets report 𝑝-values of these parameters. The in-sample period starts from

January 5, 2000, to August 6, 2004, resulting in 1,000 observations. *** denotes significance at 1% level, ** denotes sig-

nificance at 5% level, and * denotes significance at 10% level. Note that the parameters are estimated under Gaussian

distribution.


56

Table B2. – Forecast loss coefficients of HSI

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 9.043 1.212 9.255 1.244 9.293 1.338 9.305 1.119 GARCH(1,1) 3.745 0.481 3.992 0.506 4.353 0.532 4.810 0.603 GJR-GARCH(1,1) 3.795 0.487 4.078 0.511 4.446 0.542 4.909 0.615 CGARCH(1,1) 3.471*** 0.462 3.727*** 0.487 4.082*** 0.516*** 4.535*** 0.588 APARCH(1,1) 4.068 0.494 4.335 0.516 4.749 0.546 5.181 0.619 GARCHX(1,1,1) 3.859 0.451*** 4.145 0.475*** 4.493 0.516 4.951 0.581***

Panel B: RV (10-min) ARCH(1) 9.078 1.235 9.288 1.131 9.284 1.389 9.311 1.079 GARCH(1,1) 3.781 0.506 4.035 0.532 4.404 0.560 4.860 0.628 GJR-GARCH(1,1) 3.826 0.514 4.119 0.539 4.494 0.569 4.960 0.638 CGARCH(1,1) 3.509*** 0.487 3.773*** 0.513 4.137*** 0.543*** 4.587*** 0.612 APARCH(1,1) 4.090 0.522 4.369 0.545 4.797 0.574 5.227 0.642 GARCHX(1,1,1) 3.886 0.477*** 4.186 0.501*** 4.541 0.543 5.000 0.606***

Panel C: RP (Eq. 2) ARCH(1) 9.268 1.262 9.512 1.387 9.413 1.810 9.560 1.260 GARCH(1,1) 4.206 0.680 4.444 0.704 4.864 0.735 5.332 0.807 GJR-GARCH(1,1) 4.198 0.686 4.516 0.711 4.944 0.745 5.424 0.819 CGARCH(1,1) 3.975*** 0.659 4.221*** 0.683 4.626*** 0.716*** 5.084*** 0.791 APARCH(1,1) 4.401 0.693 4.736 0.716 5.229 0.749 5.672 0.821 GARCHX(1,1,1) 4.265 0.647*** 4.576 0.673*** 4.985 0.721 5.458 0.784***

Panel D: RP (Eq. 3) ARCH(1) 9.248 1.281 9.445 1.368 9.409 1.326 9.548 1.262 GARCH(1,1) 4.240 0.643 4.445 0.671 4.821 0.702 5.229 0.767 GJR-GARCH(1,1) 4.265 0.650 4.521 0.678 4.905 0.713 5.320 0.780 CGARCH(1,1) 3.998*** 0.623 4.210*** 0.650 4.572*** 0.683*** 4.974*** 0.750 APARCH(1,1) 4.500 0.658 4.760 0.683 5.194 0.716 5.577 0.782 GARCHX(1,1,1) 4.330 0.612*** 4.593 0.641*** 4.946 0.687 5.358 0.744***

Table B2. – This table displays the empirical loss coefficients of various GARCH-type models of the Hang Seng Index at multiple horizons

under different variance proxies. For the one-step, five-steps, 21-steps, and 63-steps ahead direct forecasts, we obtained 3,078, 3,074, 3,058,

and 3,016 out-of-sample observations, respectively. Note that RMSE coefficients have been scaled up for convenience sake. *** indicates the


in this sample.


57

Table B3. – Forecast loss coefficients of IPC

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 16.969 1.278 17.003 1.243 17.094 1.241 17.266 1.248 GARCH(1,1) 2.114 0.437 2.306 0.481 2.622 0.549 2.876 0.642 GJR-GARCH(1,1) 2.048 0.421 2.302 0.466 2.667 0.545 2.919 0.628 CGARCH(1,1) 1.987 0.417*** 2.168*** 0.466 2.441*** 0.535*** 2.666*** 0.625 APARCH(1,1) 1.979*** 0.418 2.232 0.464*** 2.617 0.542 2.801 0.606*** GARCHX(1,1,1) 2.797 0.421 3.069 0.492 3.407 0.562 3.635 0.642

Panel B: RV (10-min) ARCH(1) 16.922 1.252 16.948 1.216 17.049 1.214 17.223 1.226 GARCH(1,1) 2.004 0.404 2.223 0.453 2.596 0.528 2.891 0.633 GJR-GARCH(1,1) 1.906 0.390 2.216 0.441 2.639 0.529 2.930 0.620 CGARCH(1,1) 1.904 0.385*** 2.107*** 0.439*** 2.429*** 0.516*** 2.687*** 0.617 APARCH(1,1) 1.848*** 0.388 2.157 0.440 2.601 0.526 2.813 0.598*** GARCHX(1,1,1) 2.649 0.393 2.955 0.463 3.351 0.542 3.617 0.630

Panel C: RP (Eq. 2) ARCH(1) 16.868 1.235 16.962 1.249 17.016 1.277 17.171 1.242 GARCH(1,1) 2.021 0.445 2.164 0.486 2.561 0.563 2.884 0.678 GJR-GARCH(1,1) 1.938 0.430 2.174 0.475 2.582 0.563 2.940 0.663 CGARCH(1,1) 1.928 0.433 2.068*** 0.477 2.412*** 0.558*** 2.693*** 0.671 APARCH(1,1) 1.857*** 0.428*** 2.113 0.472*** 2.558 0.562 2.827 0.643*** GARCHX(1,1,1) 2.602 0.453 2.842 0.509 3.268 0.593 3.568 0.690

Panel D: RP (Eq. 3) ARCH(1) 16.857 1.238 16.986 1.244 17.012 1.260 17.196 1.250 GARCH(1,1) 1.922 0.443 2.109 0.480 2.468 0.546 2.705 0.643 GJR-GARCH(1,1) 1.875 0.434 2.116 0.472 2.510 0.545 2.769 0.632 CGARCH(1,1) 1.777*** 0.427*** 1.965*** 0.467*** 2.284*** 0.536*** 2.492*** 0.634 APARCH(1,1) 1.779 0.430 2.043 0.469 2.459 0.542 2.648 0.612*** GARCHX(1,1,1) 2.611 0.451 2.884 0.502 3.241 0.568 3.451 0.655

Table B3. – This table displays the empirical loss coefficients of various GARCH-type models of the IPC index at multiple horizons under

different variance proxies. For the one-step, five-steps, 21-steps, and 63-steps ahead direct forecasts, we obtained 3,078, 3,074, 3,058, and

3,016 out-of-sample observations, respectively. Note that RMSE coefficients have been scaled up for convenience sake. *** indicates the


in this sample.


58

Table B4. – Forecast loss coefficients of KOSPI

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 10.137 1.374 10.186 2.024 10.241 1.845 10.438 1.324 GARCH(1,1) 1.967 0.403 2.314 0.431 2.959 0.481 3.321 0.556 GJR-GARCH(1,1) 1.918 0.400 2.352 0.436 3.128 0.499 3.542 0.564 CGARCH(1,1) 1.859 0.383*** 2.222 0.413*** 2.852*** 0.467*** 3.218*** 0.544*** APARCH(1,1) 1.789*** 0.402 2.180*** 0.437 2.875 0.501 3.250 0.559 GARCHX(1,1,1) 2.770 0.427 2.925 0.451 3.509 0.500 4.051 0.568

Panel B: RV (10-min) ARCH(1) 10.116 1.408 10.189 1.838 10.245 1.750 10.429 1.274 GARCH(1,1) 2.069 0.418 2.404 0.448 2.996 0.501 3.335 0.576 GJR-GARCH(1,1) 2.004 0.414 2.467 0.452 3.178 0.519 3.558 0.583 CGARCH(1,1) 1.963 0.399*** 2.314 0.432*** 2.891*** 0.488*** 3.233*** 0.565*** APARCH(1,1) 1.869*** 0.416 2.292*** 0.453 2.926 0.521 3.267 0.577 GARCHX(1,1,1) 2.819 0.440 3.007 0.467 3.565 0.518 4.065 0.587

Panel C: RP (Eq. 2) ARCH(1) 10.200 1.383 10.230 2.096 10.396 1.760 10.600 1.409 GARCH(1,1) 2.610 0.583 2.961 0.614 3.526 0.669 3.813 0.742 GJR-GARCH(1,1) 2.358 0.574 2.963 0.616 3.698 0.689 4.011 0.749 CGARCH(1,1) 2.531 0.564*** 2.897 0.597*** 3.435*** 0.654*** 3.723*** 0.730*** APARCH(1,1) 2.319*** 0.574 2.819*** 0.616 3.484 0.692 3.754 0.742 GARCHX(1,1,1) 3.126 0.601 3.341 0.629 4.009 0.689 4.453 0.752

Panel D: RP (Eq. 3) ARCH(1) 10.305 1.298 10.238 2.372 10.405 1.929 10.621 1.406 GARCH(1,1) 2.605 0.561 2.980 0.594 3.548 0.648 3.809 0.718 GJR-GARCH(1,1) 2.398 0.554 2.960 0.597 3.722 0.670 4.013 0.729 CGARCH(1,1) 2.524 0.541*** 2.914 0.576*** 3.456*** 0.634*** 3.719*** 0.704*** APARCH(1,1) 2.374*** 0.555 2.809*** 0.598 3.500 0.673 3.758 0.724 GARCHX(1,1,1) 3.194 0.582 3.370 0.613 4.036 0.668 4.458 0.730

Table B4. – This table displays the empirical loss coefficients of various GARCH-type models of KOSPI at multiple horizons under different

variance proxies. For the one-step, five-steps, 21-steps, and 63-steps ahead direct forecasts, we obtained 3,078, 3,074, 3,058, and 3,016 out-

of-sample observations, respectively. Note that RMSE coefficients have been scaled up for convenience sake. *** indicates the preferred

forecasting model that yields the lowest loss function coefficient, and thus illustrates the best out-of-sample forecast performance in this

sample.

Table B5. – Coefficients of determination

RV (5-min) RV (10-min) RP (Eq. 2) RP (Eq. 3)

Indices 𝑹𝟐 𝑹𝟐 𝑹𝟐 𝑹𝟐

S&P 500 0.594 0.591 0.548 0.480

HSI 0.429 0.409 0.270 0.245

IPC 0.294 0.339 0.349 0.347

KOSPI 0.478 0.460 0.341 0.316

Table B5. – This matrix illustrates the adjusted 𝑅2 coefficients of the out-of-sample

regressions of various volatility proxies on the VIX variance proxy. 3,078 out-of-sam-

ple observations were used in these regressions.


59

Table B6. – Diebold-Mariano test statistics of one-day ahead forecasts of S&P 500

Forecast model RMSE QLIKE

Panel A: RV (5-min) ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX

ARCH(1) - -

GARCH(1,1) 5.02*** - 8.46*** -

GJR-GARCH(1,1) 4.96*** 3.62*** - 8.80*** 7.37*** -

CGARCH(1,1) 4.98*** 3.39*** -3.72*** - 8.96*** 2.38** -1.51 -

APARCH(1,1) 4.93*** 3.36*** 0.90 3.34*** - 9.14*** 3.91*** 0.36 2.88*** -

GARCHX(1,1,1) 4.89*** 3.53*** 3.25*** 3.57*** 3.75*** - 9.48*** 3.49*** 1.93* 3.59*** 2.14** -

Panel B: RV (10-min) ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX

ARCH(1) - -

GARCH(1,1) 4.98*** - 8.90*** -

GJR-GARCH(1,1) 4.94*** 3.65*** - 9.27*** 6.19*** -

CGARCH(1,1) 4.94*** 3.34*** -3.78*** - 9.42*** 2.32** -1.37 -

APARCH(1,1) 4.90*** 3.38*** 1.18 3.38*** - 9.63*** 4.23*** 0.37 2.92*** -

GARCHX(1,1,1) 4.84*** 3.46*** 3.03*** 3.50*** 3.49*** - 9.79*** 3.62*** 1.92* 3.77*** 2.16** -

Panel C: RP (Eq. 2) ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX

ARCH(1) - -

GARCH(1,1) 4.93*** - 13.14*** -

GJR-GARCH(1,1) 4.89*** 3.55*** - 14.50*** 5.90*** -

CGARCH(1,1) 4.89*** 3.36*** -3.59*** - 13.57*** 2.27** -1.71* -

APARCH(1,1) 4.85*** 3.24*** 1.21 3.12*** - 14.38*** 3.65*** 0.28 2.49** -

GARCHX(1,1,1) 4.76*** 3.45*** 3.34*** 3.47*** 3.57*** - 12.21*** 4.16*** 2.42** 4.36*** 2.61*** -

Panel D: RP (Eq. 3) ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX ARCH GARCH GJR-GARCH CGARCH APARCH GARCHX

ARCH(1) - -

GARCH(1,1) 5.01*** - 13.39*** -

GJR-GARCH(1,1) 4.98*** 3.50*** - 14.86*** 5.45*** -

CGARCH(1,1) 4.96*** 3.40*** -0.81 - 13.83*** 2.70*** -1.04 -

APARCH(1,1) 4.93*** 3.11*** 0.94 1.85* - 14.92*** 3.02*** 0.10 1.29 -

GARCHX(1,1,1) 4.80*** 3.45*** 3.40*** 3.47*** 3.59*** - 11.72*** 3.75*** 2.24** 3.66*** 2.39** -

Table B6. – This table illustrates the Diebold-Mariano test results of one-day ahead forecasts of S&P 500 for various GARCH-type models. As explained in Diebold (2015), this test statistic can

be calculated through regression of the loss differential on a constant with Newey-West standard errors. A positive DM test statistic implies that the column model has the tendency to generate

less accurate forecasts compared to the corresponding row model, whereas a negative DM test statistic implies that the column model tends to predict more accurately than the corresponding

row model. Statistical significance of the Diebold-Mariano test statistics are determined by the conventional two-tailed critical values of a standard normal distribution. *, **, and *** denote

significance at 10%, 5%, and 1% levels, respectively. For the one-step ahead forecasts, 3,078 out-of-sample observations were used to determine the corresponding DM test statistics.


60

Table B7. – Diebold-Mariano test statistics of one-week ahead forecasts of S&P 500



ARCH(1) - -

GARCH(1,1) 5.02*** - 9.26*** -

GJR-GARCH(1,1) 5.03*** 3.51*** - 9.48*** 2.05** -

CGARCH(1,1) 4.97*** 3.38*** 1.84* - 9.65*** 2.00** -0.92 -

APARCH(1,1) 4.97*** 3.04*** 0.90 -4.77*** - 9.52*** 2.25** 0.32 1.51 -

GARCHX(1,1,1) 4.88*** 3.39*** 3.17*** 3.39*** 3.52*** - 9.93*** 2.62*** 1.49 2.44** 2.05** -


ARCH(1) - -

GARCH(1,1) 4.98*** - 9.41*** -

GJR-GARCH(1,1) 4.99*** 3.57*** - 9.65*** 2.67*** -

CGARCH(1,1) 4.94*** 3.36*** 1.63 - 9.79*** 2.04** -1.18 -

APARCH(1,1) 4.94*** 3.09*** 0.89 -4.09*** - 9.59*** 2.24** 0.12 1.54 -

GARCHX(1,1,1) 4.85*** 3.37*** 3.09*** 3.36*** 3.47*** - 9.92*** 2.69** 1.51 2.56** 2.18** -


ARCH(1) - -

GARCH(1,1) 5.00*** - 10.83*** -

GJR-GARCH(1,1) 5.01*** 3.35*** - 11.63*** 3.13*** -

CGARCH(1,1) 4.95*** 3.38*** 2.81*** - 11.42*** 2.51** -1.36 -

APARCH(1,1) 4.96*** 2.92*** 1.34 -4.38*** - 11.42*** 2.44** -0.07 1.37 -

GARCHX(1,1,1) 4.83*** 3.37*** 3.26*** 3.36*** 3.50*** - 10.98*** 2.44** 1.41 2.23** 1.83* -


ARCH(1) - -

GARCH(1,1) 5.05*** - 12.18*** -

GJR-GARCH(1,1) 5.06*** 3.12*** - 13.72*** 3.31*** -

CGARCH(1,1) 4.99*** 3.39*** 3.10*** - 12.76*** 2.96*** -0.94 -

APARCH(1,1) 5.01*** 2.66*** 0.87 -4.15*** - 12.74*** 1.95* -0.34 0.45 -

GARCHX(1,1,1) 4.83*** 3.40*** 3.35*** 3.40*** 3.52*** - 10.08*** 2.35** 1.39 2.01** 1.84* -

Table B7. – This table illustrates the Diebold-Mariano test results of one-week ahead forecasts of S&P 500 for various GARCH-type models. As explained in Diebold (2015), this test statistic

can be calculated through regression of the loss differential on a constant with Newey-West standard errors. A positive DM test statistic implies that the column model has the tendency to

generate less accurate forecasts compared to the corresponding row model, whereas a negative DM test statistic implies that the column model tends to predict more accurately than the

corresponding row model. Statistical significance of the Diebold-Mariano test statistics are determined by the conventional two-tailed critical values of a standard normal distribution. *, **,

and *** denote significance at 10%, 5%, and 1% levels, respectively. For the one-week ahead forecasts, 3,074 out-of-sample observations were used to determine the corresponding DM test

statistics.


61

Table B8. – Diebold-Mariano test statistics of one-month ahead forecasts of S&P 500



ARCH(1) - -

GARCH(1,1) 5.16*** - 5.52*** -

GJR-GARCH(1,1) 5.18*** -2.37** - 5.22*** -2.43** -

CGARCH(1,1) 5.09*** 3.33*** 3.43*** - 4.93*** -2.54** -1.23 -

APARCH(1,1) 5.14*** -4.89*** -0.18 -3.77*** - 4.17*** -1.58 -1.36 -1.04 -

GARCHX(1,1,1) 4.92*** 3.32*** 3.36*** 3.32*** 3.46*** - 6.46*** 1.68* 2.81*** 2.56** 2.37** -


ARCH(1) - -

GARCH(1,1) 5.13*** - 5.67*** -

GJR-GARCH(1,1) 5.15*** -2.70*** - 5.35*** -2.65*** -

CGARCH(1,1) 5.06*** 3.32*** 3.40*** - 5.10*** -2.65*** -1.10 -

APARCH(1,1) 5.11*** -5.95*** -0.18 -3.72*** - 4.26*** -1.70* -1.43 -1.16 -

GARCHX(1,1,1) 4.89*** 3.30*** 3.33*** 3.29*** 3.44*** - 6.71*** 1.70* 2.81*** 2.53** 2.44** -


ARCH(1) - -

GARCH(1,1) 5.10*** - 5.12*** -

GJR-GARCH(1,1) 5.13*** -2.98*** - 4.89*** -1.88* -

CGARCH(1,1) 5.03*** 3.32*** 3.40*** - 4.77*** -2.40** -1.36 -

APARCH(1,1) 5.08*** -5.95*** -0.12 -3.76*** - 4.00*** -1.54 -1.43 -1.14 -

GARCHX(1,1,1) 4.85*** 3.32*** 3.35*** 3.32*** 3.46*** - 5.87*** 2.31** 3.20*** 3.29*** 2.52** -


ARCH(1) - -

GARCH(1,1) 5.15*** - 8.83*** -

GJR-GARCH(1,1) 5.17*** -3.65*** - 8.47*** -1.54 -

CGARCH(1,1) 5.07*** 3.35*** 3.50*** - 7.60*** -2.05** -1.35 -

APARCH(1,1) 5.12*** -5.96*** -0.19 -3.82*** - 6.14*** -1.58 -1.54 -1.13 -

GARCHX(1,1,1) 4.86*** 3.34*** 3.39*** 3.34*** 3.48*** - 9.69*** 2.89*** 3.57*** 3.55*** 2.87*** -

Table B8. – This table illustrates the Diebold-Mariano test results of one-month ahead forecasts of S&P 500 for various GARCH-type models. As explained in Diebold (2015), this test statistic




and *** denote significance at 10%, 5%, and 1% levels, respectively. For the one-month ahead forecasts, 3,058 out-of-sample observations were used to determine the corresponding DM test

statistics.


62

Table B9. – Diebold-Mariano test statistics of one-quarter ahead forecasts of S&P 500



ARCH(1) - -

GARCH(1,1) 5.23*** - 3.79*** -

GJR-GARCH(1,1) 5.27*** -3.60*** - 3.54*** -0.78 -

CGARCH(1,1) 5.15*** 3.35*** 3.44*** - 3.35*** -0.95 -0.25 -

APARCH(1,1) 5.22*** -4.39*** 0.79 -3.61*** - 3.43*** -0.57 -0.08 0.17 -

GARCHX(1,1,1) 4.89*** 3.29*** 3.32*** 3.27*** 3.38*** - 4.39*** 1.40 1.32 1.26 1.12 -


ARCH(1) - -

GARCH(1,1) 5.19*** - 4.06*** -

GJR-GARCH(1,1) 5.23*** -3.56*** - 3.78*** -0.67 -

CGARCH(1,1) 5.11*** 3.34*** 3.42*** - 3.58*** -0.88 -0.25 -

APARCH(1,1) 5.18*** -4.31*** 0.79 -3.59*** - 3.65*** -0.49 -0.05 0.19 -

GARCHX(1,1,1) 4.86*** 3.29*** 3.32*** 3.27*** 3.37*** - 4.71*** 1.47 1.26 1.26 1.06 -


ARCH(1) - -

GARCH(1,1) 5.20*** - 5.08*** -

GJR-GARCH(1,1) 5.25*** -3.63*** - 4.85*** -0.48 -

CGARCH(1,1) 5.12*** 3.33*** 3.44*** - 4.57*** -0.61 -0.14 -

APARCH(1,1) 5.19*** -4.40*** 0.67 -3.62*** - 4.64*** -0.60 -0.61 -0.34 -

GARCHX(1,1,1) 4.85*** 3.29*** 3.32*** 3.27*** 3.38*** - 6.03*** 1.56 1.33 1.30 1.33 -


ARCH(1) - -

GARCH(1,1) 5.21*** - 6.09*** -

GJR-GARCH(1,1) 5.25*** -3.77*** - 6.04*** -0.09 -

CGARCH(1,1) 5.12*** 3.34*** 3.48*** - 5.63*** -0.19 -0.09 -

APARCH(1,1) 5.19*** -4.74*** 0.68 -3.67*** - 5.82*** -0.36 -0.68 -0.35 -

GARCHX(1,1,1) 4.82*** 3.30*** 3.34*** 3.28*** 3.39*** - 7.02*** 2.08** 1.52 1.55 1.52 -

Table B9. – This table illustrates the Diebold-Mariano test results of one-quarter ahead forecasts of S&P 500 for various GARCH-type models. As explained in Diebold (2015), this test statistic




and *** denote significance at 10%, 5%, and 1% levels, respectively. For the one-quarter ahead forecasts, 3,016 out-of-sample observations were used to determine the corresponding DM test

statistics.


63

Table B10. – Results of MCS for HSI

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0 0 0 GJR-GARCH(1,1) 0 0 0 0 0 0 0 0 CGARCH(1,1) 1.000*** 0.381*** 1.000*** 0.487*** 1.000*** 1.000*** 1.000*** 0.744*** APARCH(1,1) 0 0 0 0 0 0 0 0 GARCHX(1,1,1) 0 1.000*** 0 1.000*** 0 1.000*** 0 1.000***

Panel B: RV (10-min) ARCH(1) 0 0 0 0 0 0.265*** 0 0 GARCH(1,1) 0 0 0 0 0 0 0 0 GJR-GARCH(1,1) 0 0 0 0 0 0.016 0 0 CGARCH(1,1) 1.000*** 0.480*** 1.000*** 0.506*** 1.000*** 1.000*** 1.000*** 0.738*** APARCH(1,1) 0 0 0 0 0 0.044 0 0 GARCHX(1,1,1) 0 1.000*** 0 1.000*** 0 1.000*** 0 1.000***

Panel C: RP (Eq. 2) ARCH(1) 0 0 0 0 0 0.419*** 0 0 GARCH(1,1) 0 0 0 0 0 0 0 0 GJR-GARCH(1,1) 0 0 0 0 0 0.010 0 0 CGARCH(1,1) 1.000*** 0.458*** 1.000*** 0.580*** 1.000*** 1.000*** 1.000*** 0.679*** APARCH(1,1) 0 0 0 0 0 0.035 0 0 GARCHX(1,1,1) 0 1.000*** 0 1.000*** 0 1.000*** 0 1.000***

Panel D: RP (Eq. 3) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0 0 0 GJR-GARCH(1,1) 0 0 0 0 0 0 0 0 CGARCH(1,1) 1.000*** 0.398*** 1.000*** 0.592*** 1.000*** 1.000*** 1.000*** 0.723*** APARCH(1,1) 0 0 0 0 0 0 0 0 GARCHX(1,1,1) 0 1.000*** 0 1.000*** 0 0.968*** 0 1.000***

Table B10. – This table provides the 𝑝-values of Model Confidence Sets for the Hang Seng Index forecast losses at multiple horizons. A block

bootstrap approach of 1,000 resamples, block length equal to forecast horizon, and 5% significance level are used to determine these coef-

ficients. Note that we set ℒ = 2 for the one-step ahead forecasts. *** denotes the forecast model is part of ℳ̂0.95∗ .


64

Table B11. – Results of MCS for IPC

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0.002 0 0.024 0 0.147*** GJR-GARCH(1,1) 0 0.729*** 0 0.942*** 0 0.804*** 0 0.099*** CGARCH(1,1) 1.000*** 1.000*** 1.000*** 0.999*** 1.000*** 1.000*** 1.000*** 0.811*** APARCH(1,1) 0 0.999*** 0 1.000*** 0 0.978*** 0 1.000*** GARCHX(1,1,1) 0 0.986*** 0 0.216*** 0 0.416*** 0 0.497***

Panel B: RV (10-min) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0 0 0.001 0 0.024 0 0.302*** GJR-GARCH(1,1) 0 0.918*** 0 0.996*** 0 0.701*** 0 0.211*** CGARCH(1,1) 0 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 0.864*** APARCH(1,1) 1.000*** 0.982*** 0 1.000*** 0 0.955*** 0 1.000*** GARCHX(1,1,1) 0 0.836*** 0 0.141*** 0 0.487*** 0 0.519***

Panel C: RP (Eq. 2) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0.008 0 0.082*** 0 0.638*** 0 0.600*** GJR-GARCH(1,1) 0 0.944*** 0 0.978*** 0 0.968*** 0 0.251*** CGARCH(1,1) 0 0.930*** 1.000*** 0.975*** 1.000*** 1.000*** 1.000*** 0.613*** APARCH(1,1) 1.000*** 1.000*** 0 1.000*** 0 1.000*** 0 1.000*** GARCHX(1,1,1) 0 0.296*** 0 0.067*** 0 0.205*** 0 0.155***

Panel D: RP (Eq. 3) ARCH(1) 0 0 0 0 0 0 0 0 GARCH(1,1) 0 0.002 0 0.031 0 0.038 0 0.617*** GJR-GARCH(1,1) 0 0.917*** 0 0.969*** 0 0.844*** 0 0.227*** CGARCH(1,1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 0.757*** APARCH(1,1) 0 0.993*** 0 1.000*** 0 0.986*** 0 1.000*** GARCHX(1,1,1) 0 0.234*** 0 0.065*** 0 0.168*** 0 0.221***

Table B11. – This table provides the 𝑝-values of Model Confidence Sets for the IPC forecast losses at multiple horizons. A block bootstrap




65

Table B12. – Results of MCS for KOSPI

𝒉 = 𝟏 𝒉 = 𝟓 𝒉 = 𝟐𝟏 𝒉 = 𝟔𝟑


Panel A: RV (5-min) ARCH(1) 0 0 0 0.369*** 0 0.275*** 0 0 GARCH(1,1) 0 0 0 0 0 0.004 0 0.045 GJR-GARCH(1,1) 0 0.209*** 0 0.013 0 0.045 0 0.377*** CGARCH(1,1) 0 1.000*** 0.750*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** APARCH(1,1) 1.000*** 0.306*** 1.000*** 0.050*** 0 0.077*** 0 0.776*** GARCHX(1,1,1) 0 0.209*** 0 0.369*** 0 1.000*** 0 1.000***

Panel B: RV (10-min) ARCH(1) 0 0 0 0.316*** 0 0.280*** 0 0 GARCH(1,1) 0 0 0 0 0 0.003 0 0.205*** GJR-GARCH(1,1) 0 0.333*** 0 0.085*** 0 0.057*** 0 0.451*** CGARCH(1,1) 0 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** APARCH(1,1) 1.000*** 0.385*** 0.235*** 0.135*** 0 0.098*** 0 0.862*** GARCHX(1,1,1) 0 0.333*** 0 0.316*** 0 1.000*** 0 1.000***

Panel C: RP (Eq. 2) ARCH(1) 0 0 0 0.379*** 0 0 0 0 GARCH(1,1) 0 0 0 0 0 0.001 0 0.069*** GJR-GARCH(1,1) 0 0.606*** 0 0.213*** 0 0.057*** 0 0.420*** CGARCH(1,1) 0 1.000*** 0 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** APARCH(1,1) 1.000*** 0.723*** 1.000*** 0.415*** 0 0.092*** 0 0.893*** GARCHX(1,1,1) 0 0.606*** 0 1.000*** 0 1.000*** 0 1.000***

Panel D: RP (Eq. 3) ARCH(1) 0 0 0 0.493*** 0 0.274*** 0 0 GARCH(1,1) 0 0 0 0 0 0.003 0 0.044 GJR-GARCH(1,1) 0 0.404*** 0 0.048 0 0.053*** 0 0.165*** CGARCH(1,1) 0 1.000*** 0 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** APARCH(1,1) 1.000*** 0.478*** 1.000*** 0.140*** 0 0.096*** 0 0.575*** GARCHX(1,1,1) 0 0.404*** 0 0.493*** 0 1.000*** 0 1.000***

Table B12. – This table provides the 𝑝-values of Model Confidence Sets for the KOSPI forecast losses at multiple horizons. A block bootstrap




66

Section C – Supplementary graphs

Figure C1. – Annualised volatility proxies of HSI index

Figure C1. – This figure displays volatility estimates of the HSI index based on four different proxies, namely the 5-minute and

10-minute RV, and the two range-based variances based on Eq. (2) and Eq. (3). The volatility estimates are annualised, using

the following formula from Patton (2011a): 𝜎𝑡𝐴 = √252 × 𝑉𝐴�̂�𝑡. Furthermore, the vertical axes in all panels are scaled up by

1/100. The sample period starts from January 5, 2000, to March 27, 2018.


67

Figure C2. – Annualised volatility proxies of IPC index

Figure C2. – This figure displays volatility estimates of the IPC index based on four different proxies, namely the 5-minute and

10-minute RV, and the two range-based variances based on Eq. (2) and Eq. (3). The volatility estimates are annualised, using

the following formula from Patton (2011a): 𝜎𝑡𝐴 = √252 × 𝑉𝐴�̂�𝑡. Furthermore, the vertical axes in all panels are scaled up by

1/100. The sample period starts from January 5, 2000, to March 27, 2018.


68

Figure C3. – Annualised volatility proxies of KOSPI index

Figure C3. – This figure displays volatility estimates of the KOSPI index based on four different proxies, namely the 5-minute

and 10-minute RV, and the two range-based variances based on Eq. (2) and Eq. (3). The volatility estimates are annualised,

using the following formula from Patton (2011a): 𝜎𝑡𝐴 = √252 × 𝑉𝐴�̂�𝑡. Furthermore, the vertical axes in all panels are scaled

up by 1/100. The sample period starts from January 5, 2000, to March 27, 2018.


69

Figure C4. – Autocorrelations of HSI volatility proxies

Figure C4. – In this figure, correlograms of the four different volatility proxies of the HSI index are demonstrated in the above

panels. A window of 500 lags is used in all four panels. Furthermore, the light grey dashes represent the Bartlett’s formula for

𝑀𝐴(𝑞) 95% confidence bands, i.e. 5% significance limits.


70

Figure C5. – Autocorrelations of IPC volatility proxies

Figure C5. – In this figure, correlograms of the four different volatility proxies of the IPC index are demonstrated in the above




71

Figure C6. – Autocorrelations of KOSPI volatility proxies

Figure C6. – In this figure, correlograms of the four different volatility proxies of the KOSPI index are demonstrated in the above




72

Figure C7. – Partial autocorrelations of HSI volatility proxies

Figure C7. – In this figure, partial autocorrelations of the four different volatility proxies of the HSI index are demonstrated in the

above panels. A window of 40 lags is used in all four panels. Furthermore, the light grey dashes represent the 95% confidence

bands, which can be calculated as 𝑠𝑒 = 1 √𝑛⁄ .


73

Figure C8. – Partial autocorrelations of IPC volatility proxies

Figure C8. – In this figure, partial autocorrelations of the four different volatility proxies of the IPC index are demonstrated in

the above panels. A window of 40 lags is used in all four panels. Furthermore, the light grey dashes represent the 95% confidence

bands, which can be calculated as 𝑠𝑒 = 1 √𝑛⁄ .


74

Figure C9. – Partial autocorrelations of KOSPI volatility proxies

Figure C9. – In this figure, partial autocorrelations of the four different volatility proxies of the KOSPI index are demonstrated

in the above panels. A window of 40 lags is used in all four panels. Furthermore, the light grey dashes represent the 95% confi-

dence bands, which can be calculated as 𝑠𝑒 = 1 √𝑛⁄ .


75

Figure C10. – HSI index

Figure C10. – This figure illustrates the historical data of the HSI index price level, log returns, and squared returns. These are displayed in Panels

A, B, and C, respectively. The sample period starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.


76

Figure C11. – IPC index

Figure C11. – This figure illustrates the historical data of the IPC index price level, log returns, and squared returns. These are displayed in Panels

A, B, and C, respectively. The sample period starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.


77

Figure C12. – KOSPI index

Figure C12. – This figure illustrates the historical data of the KOSPI index price level, log returns, and squared returns. These are displayed in

Panels A, B, and C, respectively. The sample period starts from January 5, 2000, to March 27, 2018, which contains 4,078 observations.


78

Figure C13. – Historical behaviour of variance proxies of S&P 500

Figure C13. – This graph plot illustrates the historical behaviour of the variance proxies of the S&P 500. We observe that the VIX

variance proxy, which is calculated as 휁𝑡 = (1 252⁄ )(𝑉𝐼𝑋𝑡 100⁄ )2, tracks other variance proxies rather well. The sample period starts

from January 5, 2000, to March 27, 2018, which contains 4,078 observations.

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